#set document(title: "6.5 Quadratic Inequalities", author: "OpenStax / XYZ Homework")
#set page(width: 8.5in, height: auto, margin: 1in)
#import "@preview/cetz:0.5.2"
#set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en")
#show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math"))
#set par(justify: true, leading: 0.62em, spacing: 0.9em)
#set enum(spacing: 1.1em)   // room between list items so tall inline fractions don't collide
#set list(spacing: 1.1em)
#set table(stroke: 0.5pt + rgb("#c7ccd3"))

#let BLUE = rgb("#183B6F")     // brand navy — section bars + example/solution labels (white on navy 11.09:1)
#let ORANGE = rgb("#A94509")   // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text)
#let RED = rgb("#DC2626")      // brand error-600
#let GREEN = rgb("#059669")    // brand success-600 (decoration only; small green text uses green-text #007942)

#show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt,
  fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt,
  text(fill: white, weight: "bold", size: 19pt, it.body))
#show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt,
  fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt,
  text(fill: white, weight: "bold", size: 12pt, it.body))
#show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body)
#show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body)

#let examplebox(label, title, body) = block(width: 100%, breakable: true,
  fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt,
  above: 12pt, below: 12pt)[
  #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt,
    text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]]
  #body]

// rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade
#let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint,
    stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt),
    above: 11pt, below: 11pt)[
    #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)]
    #linebreak() #body]

#let solutionbox(body) = block(above: 4pt, below: 8pt)[
  #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body]

#let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"),
  stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[
  #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))]

// Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a
// consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the
// print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a
// degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero.
#let _STD_W = 3.5
#let _WIDE_W = 5.6
#let _MAX_H = 3.4
#let _ASPECT_WIDE = 2.2
#let _UPSCALE_MAX = 1.15
#let stdfig(body) = context {
  let m = measure(body)
  let w = calc.max(m.width / 1in, 0.01)
  let h = calc.max(m.height / 1in, 0.01)
  let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W }
  let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX)
  align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body)))
}

#show figure: set block(breakable: false)
#set figure(gap: 8pt)
#show figure.caption: set text(size: 8.5pt, fill: rgb("#555"))


    
== 6.5#h(0.6em)Quadratic Inequalities


    
      
=== Solving Inequalities Graphically


      In Functions and Their Graphs, we used graphs to solve equations and inequalities. The graphing technique is especially helpful for solving quadratic inequalities.


      #examplebox("Example 1")[][
The Chamber of Commerce in River City plans to put on a Fourth of July fireworks display. City regulations require that fireworks at public gatherings explode higher than #math.equation(block: false, alt: "800")[$800$] feet above the ground. The mayor particularly wants to include the Freedom Starburst model, which is launched from the ground. Its height after t seconds is given by


          #math.equation(block: true, alt: "h equals f open parenthesis t close parenthesis equals 256 t minus 16 t squared")[$h = f ( t ) = 256 t − 16 t^(2)$]
          When should the Starburst explode in order to satisfy the safety regulation?


        
        #solutionbox[
We can get an approximate answer to this question by looking at the graph of the rocket's height, shown below.


          #figure(figph[graph of fireworks height], alt: "graph of fireworks height", caption: none)


          When is the rocket's height greater than 800 feet, or, in mathematical terms, for what values of #math.equation(block: false, alt: "t")[$t$] is #math.equation(block: false, alt: "h greater than 800")[$h > 800$]? The answer to this question is the solution of the inequality


          #math.equation(block: true, alt: "256 t minus 16 t squared greater than 800")[$256 t − 16 t^(2) > 800$]
          Points on the graph with #math.equation(block: false, alt: "h greater than 800")[$h > 800$] are shown in color, and the #math.equation(block: false, alt: "t")[$t$]-coordinates of those points are marked on the horizontal axis. If the Freedom Starburst explodes at any of these times, it will satisfy the safety regulation.


          From the graph, the safe time interval runs from approximately #math.equation(block: false, alt: "4.25")[$4.25$] seconds to #math.equation(block: false, alt: "11.75")[$11.75$] seconds after launch. The solution of the inequality is the set of all #math.equation(block: false, alt: "t")[$t$]-values greater than #math.equation(block: false, alt: "4.25")[$4.25$] but less than #math.equation(block: false, alt: "11.75")[$11.75$].
]
]


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The solution set in Example is called a #strong[compound inequality], because it involves more than one inequality symbol. We write this set as


        #math.equation(block: true, alt: "4.25 less than t less than 11.75")[$4.25 < t < 11.75$]
        and read "#math.equation(block: false, alt: "t")[$t$] greater than #math.equation(block: false, alt: "4.25")[$4.25$] but less than #math.equation(block: false, alt: "11.75")[$11.75$]."
]


      
        What are the solutions of the inequality #math.equation(block: false, alt: "x squared plus 1 less than 0")[$x^(2) + 1 < 0$] ?


        \_\_\_\_\_


        #solutionbox[
No solution
]


      
      
        What are the solutions of the inequality #math.equation(block: false, alt: "x squared plus 1 less than 0")[$x^(2) + 1 < 0$] ?


        + #math.equation(block: false, alt: "open parenthesis minus 1 , 1 close parenthesis")[$( − 1 , 1 )$]
+ #math.equation(block: false, alt: "open parenthesis minus ∞ , minus 1 close parenthesis union open parenthesis 1 , ∞ close parenthesis")[$( − upright(∞) , − 1 ) ∪ ( 1 , upright(∞) )$]
+ All real numbers
+ No solution


      
      #notebox("Technology", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Solving an Inequality With a Graphing Calculator]


        You can use your graphing calculator to solve the problem in Example. Graph the two functions


        #math.equation(block: true, alt: "Y sub 1, equals 256 X minus 16 X squared; Y sub 2, equals 800")[$Y_(1) & = 256 X − 16 X^(2) \ Y_(2) & = 800$]
        on the same screen. Use WINDOW settings to match the graph in Example.


        #figure(figph[GCgraph of fireworks height], alt: "GCgraph of fireworks height", caption: none)


        Then use the intersect feature to find the #math.equation(block: false, alt: "x")[$x$]-coordinates of the points where the two graphs intersect (there are two of them). These points will have #math.equation(block: false, alt: "y")[$y$]-coordinates of #math.equation(block: false, alt: "800")[$800$]. The parabola is above the line, so #math.equation(block: false, alt: "h greater than 800")[$h > 800$] when #math.equation(block: false, alt: "t")[$t$] is between these two #math.equation(block: false, alt: "x")[$x$]-values. To two decimal places, you can see that #math.equation(block: false, alt: "h greater than 800")[$h > 800$] when #math.equation(block: false, alt: "4.26 less than t less than 11.74")[$4.26 < t < 11.74$].
]


      
        + Graph the function #math.equation(block: false, alt: "y equals x squared minus 2 x minus 9")[$y = x^(2) − 2 x − 9$] in the window
                #math.equation(block: true, alt: "Xmin, equals minus 0.4, Xmax equals 9.4; Ymin, equals minus 10, Ymax equals 10")[$"Xmin" & = − 0.4 &  & "Xmax" = 9.4 \ "Ymin" & = − 10 &  & "Ymax" = 10$]
+ Use the graph to solve the inequality #math.equation(block: false, alt: "x squared minus 2 x minus 9 greater than or equal to 6")[$x^(2) − 2 x − 9 ≥ 6$]. #linebreak() Solution: \_\_\_\_\_


        You may use inequality symbols or enter your answer using interval notation. When using inequality symbols, enter "\<=" for #math.equation(block: false, alt: "less than or equal to")[$≤$], and enter "\>=" for #math.equation(block: false, alt: "greater than or equal to")[$≥$]. When using interval notation, use "inf" for #math.equation(block: false, alt: "∞")[$upright(∞)$] and use "U" (an upper case letter u) for the union symbol #math.equation(block: false, alt: "union")[$∪$].


        #solutionbox[
+ A graph is below.
+ #math.equation(block: false, alt: "x less than or equal to minus 3")[$x ≤ − 3$] or #math.equation(block: false, alt: "x greater than or equal to 5")[$x ≥ 5$]
]


        A graph for part (a):


        #figure(figph[parabola for quadratic inequality], alt: "parabola for quadratic inequality", caption: none)


      
      
        + Graph the function #math.equation(block: false, alt: "y equals x squared minus 2 x minus 9")[$y = x^(2) − 2 x − 9$] in the window
                #math.equation(block: true, alt: "Xmin, equals minus 0.4, Xmax equals 9.4; Ymin, equals minus 10, Ymax equals 10")[$"Xmin" & = − 0.4 &  & "Xmax" = 9.4 \ "Ymin" & = − 10 &  & "Ymax" = 10$]
+ Use the graph to solve the inequality #math.equation(block: false, alt: "x squared minus 2 x minus 9 greater than or equal to 6")[$x^(2) − 2 x − 9 ≥ 6$].


        #solutionbox[
+ #figure(figph[parabola for quadratic inequality], alt: "parabola for quadratic inequality", caption: none)
+ #math.equation(block: false, alt: "x less than or equal to minus 3")[$x ≤ − 3$] or #math.equation(block: false, alt: "x greater than or equal to 5")[$x ≥ 5$]
]


      
      In Example, we solved the inequality #math.equation(block: false, alt: "256 t minus 16 t squared greater than 800")[$256 t − 16 t^(2) > 800$] by comparing points on the graph of #math.equation(block: false, alt: "h equals 256 minus 16 t squared")[$h = 256 − 16 t^(2)$] with points on the line #math.equation(block: false, alt: "h equals 800")[$h = 800$]. If one side of an inequality is zero, we can compare points on the graph with the line #math.equation(block: false, alt: "y equals 0")[$y = 0$], which is the #math.equation(block: false, alt: "x")[$x$]-axis.


      #examplebox("Example 2")[][
Consider the graph of #math.equation(block: false, alt: "y equals x squared minus 4")[$y = x^(2) − 4$]. Find the solutions of the following equations and inequalities.


          + #math.equation(block: false, alt: "x squared minus 4 equals 0")[$x^(2) − 4 = 0$]
+ #math.equation(block: false, alt: "x squared minus 4 less than 0")[$x^(2) − 4 < 0$]
+ #math.equation(block: false, alt: "x squared minus 4 greater than 0")[$x^(2) − 4 > 0$]


        
        #solutionbox[
Look at the graph of #math.equation(block: false, alt: "y equals x squared minus 4")[$y = x^(2) − 4$] shown below. When we substitute a value of #math.equation(block: false, alt: "x")[$x$] into the expression #math.equation(block: false, alt: "x squared minus 4")[$x^(2) − 4$], the result is either positive, negative, or zero.


          (You can see this more clearly if you compute a few values yourself to complete the table below. Your table should agree with the coordinates of points on the graph.)


          #figure(table(
  columns: 8, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "3")[$3$]]),
  [#math.equation(block: false, alt: "y")[$y$]], [$#hide($b l a$)$], [$#hide($b l a$)$], [$#hide($b l a$)$], [$#hide($b l a$)$], [$#hide($b l a$)$], [$#hide($b l a$)$], [$#hide($b l a$)$],
))


          We will use the graph to solve the given equation and inequalities.


          #figure(figph[parabola dividing x-axis into 3 regions], alt: "parabola dividing x-axis into 3 regions", caption: none)


          + First, locate the two points on the graph where #math.equation(block: false, alt: "y equals 0")[$y = 0$]. These points are #math.equation(block: false, alt: "open parenthesis minus 2 , 0 close parenthesis")[$( − 2 , 0 )$] and #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$]. Their #math.equation(block: false, alt: "x")[$x$]-coordinates, #math.equation(block: false, alt: "minus 2")[$− 2$] and #math.equation(block: false, alt: "2")[$2$], are the solutions of the equation #math.equation(block: false, alt: "x squared minus 4 equals 0")[$x^(2) − 4 = 0$]. The two points divide the #math.equation(block: false, alt: "x")[$x$]-axis into three sections, which are labeled on the graph. On each of these sections, the value of #math.equation(block: false, alt: "x squared minus 4")[$x^(2) − 4$] is either always positive or always negative.
+ To solve #math.equation(block: false, alt: "x squared minus 4 less than 0")[$x^(2) − 4 < 0$], find the points on the graph where #math.equation(block: false, alt: "y less than 0")[$y < 0$], that is, below the #math.equation(block: false, alt: "x")[$x$]-axis. These points have #math.equation(block: false, alt: "x")[$x$]-coordinates between #math.equation(block: false, alt: "minus 2")[$− 2$] and #math.equation(block: false, alt: "2")[$2$] (labeled section II on the figure). Thus, the solution to the inequality #math.equation(block: false, alt: "x squared minus 4 less than 0")[$x^(2) − 4 < 0$] is #math.equation(block: false, alt: "minus 2 less than x less than 2")[$− 2 < x < 2$].
+ To solve #math.equation(block: false, alt: "x squared minus 4 greater than 0")[$x^(2) − 4 > 0$], locate the points on the graph where #math.equation(block: false, alt: "y greater than 0")[$y > 0$], or above the #math.equation(block: false, alt: "x")[$x$]-axis. Points with positive #math.equation(block: false, alt: "y")[$y$]-values correspond to two sections of the #math.equation(block: false, alt: "x")[$x$]-axis, labeled I and III on the figure. In section I, #math.equation(block: false, alt: "x less than minus 2")[$x < − 2$], and in section III, #math.equation(block: false, alt: "x greater than 2")[$x > 2$]. Thus, the solution to the inequality #math.equation(block: false, alt: "x squared minus 4 greater than 0")[$x^(2) − 4 > 0$] includes all values of #math.equation(block: false, alt: "x")[$x$] for which either #math.equation(block: false, alt: "x less than minus 2")[$x < − 2$] or #math.equation(block: false, alt: "x greater than 2")[$x > 2$].
]
]


      #notebox("Caution", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
In Example, the solution of the inequality #math.equation(block: false, alt: "x squared minus 4 greater than 0")[$x^(2) − 4 > 0$] is the set


        #math.equation(block: true, alt: "x less than minus 2 or x greater than 2.")[$x < − 2 " " " " " " " or " " " " " " " x > 2.$]
        This set is another type of compound inequality, and its graph consists of two pieces, as shown below.


        #figure(figph[number line with two infinite subintervals], alt: "number line with two infinite subintervals", caption: none)


        The left piece of the set is #math.equation(block: false, alt: "x less than minus 2")[$x < − 2$], and the right piece is #math.equation(block: false, alt: "x greater than 2")[$x > 2$]. It would be incorrect to describe the solution set as #math.equation(block: false, alt: "minus 2 greater than x greater than 2")[$− 2 > x > 2$], because this notation implies that #math.equation(block: false, alt: "minus 2 greater than 2")[$− 2 > 2$]. We must write the solution as two parts: #math.equation(block: false, alt: "x less than minus 2")[$x < − 2$] or #math.equation(block: false, alt: "x greater than 2")[$x > 2$].
]


      
        Use a graph of #math.equation(block: false, alt: "y equals 6 minus x squared")[$y = 6 − x^(2)$] to solve the inequalities.


        + #math.equation(block: false, alt: "6 minus x squared greater than 0")[$6 − x^(2) > 0$] #linebreak() Solution: \_\_\_\_\_
+ #math.equation(block: false, alt: "6 minus x squared less than or equal to 0")[$6 − x^(2) ≤ 0$] #linebreak() Solution: \_\_\_\_\_


        You may use inequality symbols or enter your answer using interval notation. When using inequality symbols, enter "\<=" for #math.equation(block: false, alt: "less than or equal to")[$≤$], and enter "\>=" for #math.equation(block: false, alt: "greater than or equal to")[$≥$]. When using interval notation, use "inf" for #math.equation(block: false, alt: "∞")[$upright(∞)$] and use "U" (an upper case letter u) for the union symbol #math.equation(block: false, alt: "union")[$∪$].


        #solutionbox[
+ #math.equation(block: false, alt: "minus the square root of 6 less than x less than the square root of 6")[$− sqrt(6) < x < sqrt(6)$]
+ #math.equation(block: false, alt: "x less than or equal to minus the square root of 6")[$x ≤ − sqrt(6)$] or #math.equation(block: false, alt: "x greater than or equal to the square root of 6")[$x ≥ sqrt(6)$]
]


      
      
        Use a graph of #math.equation(block: false, alt: "y equals 6 minus x squared")[$y = 6 − x^(2)$] to solve the inequalities.


        + #math.equation(block: false, alt: "6 minus x squared greater than 0")[$6 − x^(2) > 0$]
+ #math.equation(block: false, alt: "6 minus x squared less than or equal to 0")[$6 − x^(2) ≤ 0$]


        #solutionbox[
+ #math.equation(block: false, alt: "minus the square root of 6 less than x less than the square root of 6")[$− sqrt(6) < x < sqrt(6)$]
+ #math.equation(block: false, alt: "x less than or equal to minus the square root of 6")[$x ≤ − sqrt(6)$] or #math.equation(block: false, alt: "x greater than or equal to the square root of 6")[$x ≥ sqrt(6)$]
]


      
      
        Which points on the parabola #math.equation(block: false, alt: "y equals a x squared plus b x plus c")[$y = a x^(2) + b x + c$] do we need to know to solve the inequality #math.equation(block: false, alt: "a x squared plus b x plus c greater than 0")[$a x^(2) + b x + c > 0$] ?


        \_\_\_\_\_


        #solutionbox[
The #math.equation(block: false, alt: "x")[$x$]-intercepts
]


      
      
        Which points on the parabola #math.equation(block: false, alt: "y equals a x squared plus b x plus c")[$" " y = a x^(2) + b x + c " "$] do we need to know to solve the inequality #math.equation(block: false, alt: "a x squared plus b x plus c greater than 0")[$" " a x^(2) + b x + c > 0$] ?


        + The #math.equation(block: false, alt: "x")[$x$]-intercepts
+ The #math.equation(block: false, alt: "y")[$y$]-intercept
+ The vertex
+ All of these


      
      Because it is relatively easy to decide whether the #math.equation(block: false, alt: "y")[$y$]-coordinate of a point on a graph is positive or negative (the point lies above the #math.equation(block: false, alt: "x")[$x$]-axis or below the x-axis), we often rewrite a given inequality so that one side is zero.


      #examplebox("Example 3")[][
Use a graph to solve #math.equation(block: false, alt: "x squared minus 2 x minus 3 less than or equal to 12")[$" " x^(2) − 2 x − 3 ≤ 12$]


        
        #solutionbox[
We first write the inequality with zero on one side:


          #math.equation(block: true, alt: "x squared minus 2 x minus 15 less than or equal to 0")[$x^(2) − 2 x − 15 ≤ 0$]
          . We would like to find points on the graph of #math.equation(block: false, alt: "y equals x squared minus 2 x minus 15")[$y = x^(2) − 2 x − 15$] that have #math.equation(block: false, alt: "y")[$y$]-coordinates less than or equal to zero. A graph of the equation is shown below.


          You can check that the #math.equation(block: false, alt: "x")[$x$]-intercepts of the graph are #math.equation(block: false, alt: "minus 3")[$− 3$] and #math.equation(block: false, alt: "5")[$5$]. The points shown in red on the graph lie below the #math.equation(block: false, alt: "x")[$x$]-axis and have #math.equation(block: false, alt: "y less than or equal to 0")[$y ≤ 0$], so the #math.equation(block: false, alt: "x")[$x$]-coordinates of these points are the solutions of the inequality. All of these points have #math.equation(block: false, alt: "x")[$x$]-coordinates between #math.equation(block: false, alt: "minus 3")[$− 3$] and #math.equation(block: false, alt: "5")[$5$]. Thus, the solution is #math.equation(block: false, alt: "minus 3 less than or equal to x less than or equal to 5")[$− 3 ≤ x ≤ 5$], or in interval notation, #math.equation(block: false, alt: "open bracket minus 3 , 5 close bracket")[$[ − 3 , 5 ]$].


          #figure(figph[parabola dividing the x-axis into three regions], alt: "parabola dividing the x-axis into three regions", caption: none)
]
]


      
        Follow the steps below to solve the inequality #math.equation(block: false, alt: "36 plus 6 x minus x squared less than or equal to 20")[$36 + 6 x − x^(2) ≤ 20$].


        + Rewrite the inequality so that the right side is zero. #linebreak() \_\_\_\_\_#math.equation(block: false, alt: "less than or equal to 0")[$≤ 0$]
+ Graph the equation #math.equation(block: false, alt: "y equals 16 plus 6 x minus x squared")[$y = 16 + 6 x − x^(2)$].
+ Locate the points on the graph with #math.equation(block: false, alt: "y")[$y$]-coordinate less than zero, and mark the #math.equation(block: false, alt: "x")[$x$]-coordinates of the points on the #math.equation(block: false, alt: "x")[$x$]-axis.
+ Write the solution with interval notation. #linebreak() Solution: \_\_\_\_\_


        When using interval notation, use "inf" for #math.equation(block: false, alt: "∞")[$upright(∞)$] and use "U" (an upper case letter u) for the union symbol #math.equation(block: false, alt: "union")[$∪$].


        #solutionbox[
+ #math.equation(block: false, alt: "16 plus 6 x minus x squared less than or equal to 0")[$16 + 6 x − x^(2) ≤ 0$]
+ A graph is below.
+ See graph
+ #math.equation(block: false, alt: "open parenthesis minus ∞ , minus 2 close bracket union open bracket 8 , ∞ close parenthesis")[$( − upright(∞) , − 2 ] ∪ [ 8 , upright(∞) )$]
]


        A graph for parts (b) and (c):


        #figure(figph[parabola for quadratic inequality], alt: "parabola for quadratic inequality", caption: none)


      
      
        Follow the steps below to solve the inequality #math.equation(block: false, alt: "36 plus 6 x minus x squared less than or equal to 20")[$" " 36 + 6 x − x^(2) ≤ 20$].


        + Rewrite the inequality so that the right side is zero.
+ Graph the equation #math.equation(block: false, alt: "y equals 16 plus 6 x minus x squared")[$y = 16 + 6 x − x^(2)$].
+ Locate the points on the graph with #math.equation(block: false, alt: "y")[$y$]-coordinate less than zero, and mark the #math.equation(block: false, alt: "x")[$x$]-coordinates of the points on the #math.equation(block: false, alt: "x")[$x$]-axis.
+ Write the solution with interval notation.


        #solutionbox[
+ #math.equation(block: false, alt: "16 plus 6 x minus x squared less than or equal to 0")[$16 + 6 x − x^(2) ≤ 0$]
+ #figure(figph[parabola for quadratic inequality], alt: "parabola for quadratic inequality", caption: none)
+ See graph
+ #math.equation(block: false, alt: "open parenthesis minus ∞ , minus 2 close bracket union open bracket 8 , ∞ close parenthesis")[$( − upright(∞) , − 2 ] ∪ [ 8 , upright(∞) )$]
]


      
      
        Explain why you cannot write the solutions to #math.equation(block: false, alt: "x squared minus 4 greater than 0")[$x^(2) − 4 > 0$] as a single inequality.


        \_\_\_\_\_


      
      
        Explain why you cannot write the solutions to #math.equation(block: false, alt: "x squared minus 4 greater than 0")[$x^(2) − 4 > 0$] as a single inequality.


      
    
    
      
=== Solving Quadratic Inequalities Algebraically


      Although a graph is very helpful in solving inequalities, it is not completely necessary. Every quadratic inequality can be put into one of the forms


      #math.equation(block: true, alt: "a x squared plus b x plus c, less than 0 ,, a x squared plus b x plus c, greater than 0; a x squared plus b x plus c, less than or equal to 0 ,, a x squared plus b x plus c, greater than or equal to 0")[$a x^(2) + b x + c & < 0 "," & #hide($b l a n k$) a x^(2) + b x + c & > 0 \ a x^(2) + b x + c & ≤ 0 "," & #hide($b l a n k$) a x^(2) + b x + c & ≥ 0$]
      All we really need to know is whether the corresponding parabola #math.equation(block: false, alt: "y equals a x squared plus b x plus c")[$y = a x^(2) + b x + c$] opens upward or downward. Consider the parabolas shown below.


      #figure(figph[parabola opens up, the other opens down], alt: "parabola opens up, the other opens down", caption: none)


      The parabola in figure (a) opens upward. It crosses the #math.equation(block: false, alt: "x")[$x$]-axis at two points, #math.equation(block: false, alt: "x equals r sub 1")[$x = r_(1)$] and #math.equation(block: false, alt: "x equals r sub 2")[$x = r_(2)$]. At these points, #math.equation(block: false, alt: "y equals 0")[$y = 0$].


      - The graph lies below the #math.equation(block: false, alt: "x")[$x$]-axis between #math.equation(block: false, alt: "r sub 1")[$r_(1)$] and #math.equation(block: false, alt: "r sub 2")[$r_(2)$], so the solutions to the inequality #math.equation(block: false, alt: "y less than 0")[$y < 0$] lie between #math.equation(block: false, alt: "r sub 1")[$r_(1)$] and #math.equation(block: false, alt: "r sub 2")[$r_(2)$].
- The graph lies above the #math.equation(block: false, alt: "x")[$x$]-axis for #math.equation(block: false, alt: "x")[$x$]-values less than #math.equation(block: false, alt: "r sub 1")[$r_(1)$] or greater than #math.equation(block: false, alt: "r sub 2")[$r_(2)$], so the solutions to the inequality #math.equation(block: false, alt: "y greater than 0")[$y > 0$] are #math.equation(block: false, alt: "x less than r sub 1")[$x < r_(1)$] or #math.equation(block: false, alt: "x greater than r sub 2")[$x > r_(2)$].


      If the parabola opens downward, as in figure (b), the situation is reversed. The solutions to the inequality #math.equation(block: false, alt: "y greater than 0")[$y > 0$] lie between the #math.equation(block: false, alt: "x")[$x$]-intercepts, and the solutions to #math.equation(block: false, alt: "y less than 0")[$y < 0$] lie outside the #math.equation(block: false, alt: "x")[$x$]-intercepts.


      From the graphs, we see that the #math.equation(block: false, alt: "x")[$x$]-intercepts are the boundary points between the portions of the graph with positive #math.equation(block: false, alt: "y")[$y$]-coordinates and the portions with negative #math.equation(block: false, alt: "y")[$y$]-coordinates. To solve a quadratic inequality, we need only locate the #math.equation(block: false, alt: "x")[$x$]-intercepts of the corresponding graph and then decide which intervals of the #math.equation(block: false, alt: "x")[$x$]-axis produce the correct sign for #math.equation(block: false, alt: "y")[$y$].


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[To Solve a Quadratic Inequality Algebraically:]


        + Write the inequality in standard form: One side is #math.equation(block: false, alt: "0")[$0$], and the other has the form #math.equation(block: false, alt: "a x squared plus b x plus c")[$a x^(2) + b x + c$].
+ Find the #math.equation(block: false, alt: "x")[$x$]-intercepts of the graph of #math.equation(block: false, alt: "y equals a x squared plus b x plus c")[$y = a x^(2) + b x + c$] by setting #math.equation(block: false, alt: "y equals 0")[$y = 0$] and solving for #math.equation(block: false, alt: "x")[$x$].
+ Make a rough sketch of the graph, using the sign of #math.equation(block: false, alt: "a")[$a$] to determine whether the parabola opens upward or downward.
+ Decide which intervals on the #math.equation(block: false, alt: "x")[$x$]-axis give the correct sign for #math.equation(block: false, alt: "y")[$y$].
]


      #examplebox("Example 4")[][
Solve the inequality #math.equation(block: false, alt: "36 plus 6 x minus x squared less than or equal to 20")[$" " 36 + 6 x − x^(2) ≤ 20 " "$] algebraically.


        
        #solutionbox[
+ We subtract #math.equation(block: false, alt: "20")[$20$] from both sides of the inequality so that we have #math.equation(block: false, alt: "0")[$0$] on the right side.
				#math.equation(block: true, alt: "16 plus 6 x minus x squared less than or equal to 0")[$16 + 6 x − x^(2) ≤ 0$]
+ Consider the equation #math.equation(block: false, alt: "y equals 16 plus 6 x minus x squared")[$y = 16 + 6 x − x^(2)$]. To locate the #math.equation(block: false, alt: "x")[$x$]-intercepts, we set #math.equation(block: false, alt: "y equals 0")[$y = 0$] and solve for #math.equation(block: false, alt: "x")[$x$].
		#math.equation(block: true, alt: "16 plus 6 x minus x squared, equals 0, Multiply each term by minus 1.; x squared minus 6 x minus 16, equals 0, Factor the left side.; open parenthesis x minus 8 close parenthesis open parenthesis x plus 2 close parenthesis, equals 0, Apply the zero-factor principle.; x minus 8 equals 0 or x plus 2, equals 0; x equals 8 or x, equals minus 2")[$16 + 6 x − x^(2) & = 0 &  & "Multiply each term by " − 1. \ x^(2) − 6 x − 16 & = 0 &  & "Factor the left side." \ ( x − 8 ) ( x + 2 ) & = 0 &  & "Apply the zero-factor principle." \ x − 8 = 0 " " " " " " " or " " " " " " " x + 2 & = 0 \ x = 8 " " " " " " " or " " " " " " " x & = − 2$]
	The #math.equation(block: false, alt: "x")[$x$]-intercepts are #math.equation(block: false, alt: "x equals minus 2")[$x = − 2$] and #math.equation(block: false, alt: "x equals 8")[$x = 8$].
+ Make a rough sketch of the graph of #math.equation(block: false, alt: "y equals 16 plus 6 x minus x squared")[$y = 16 + 6 x − x^(2)$], as shown below. Because #math.equation(block: false, alt: "a equals minus 1 less than 0")[$a = − 1 < 0$], the graph is a parabola that opens downward.#figure(figph[parabola opening down dividing x-axis into 3 parts], alt: "parabola opening down dividing x-axis into 3 parts", caption: none)
+ We are interested in points on the graph for which #math.equation(block: false, alt: "y less than or equal to 0")[$y ≤ 0$]. The points with negative #math.equation(block: false, alt: "y")[$y$]-coordinates (that is, points below the #math.equation(block: false, alt: "x")[$x$]-axis) lie outside the #math.equation(block: false, alt: "x")[$x$]-intercepts of the graph, so the solution of the inequality is #math.equation(block: false, alt: "x less than or equal to minus 2")[$x ≤ − 2$] or #math.equation(block: false, alt: "x greater than or equal to 8")[$x ≥ 8$]. Or, using interval notation, the solution is #math.equation(block: false, alt: "open parenthesis minus ∞ , minus 2 close bracket union open bracket 8 , ∞ close parenthesis")[$( − upright(∞) , − 2 ] ∪ [ 8 , upright(∞) )$].
]
]


      
        Why do we need to know whether tha parabola in QuickCheck 2 opens up or down?


        \_\_\_\_\_


        #solutionbox[
To decide whether the solutions lie between the #math.equation(block: false, alt: "x")[$x$]-intercepts or outside them.
]


      
      
        Why do we need to know whether tha parabola in QuickCheck 2 opens up or down?


        + To decide whether to use #math.equation(block: false, alt: "greater than")[$>$] or #math.equation(block: false, alt: "less than")[$<$] in the solution.
+ To decide whether the solutions lie between the #math.equation(block: false, alt: "x")[$x$]-intercepts or outside them.
+ To decide whether the solutions are positive or negative.
+ To help us find the #math.equation(block: false, alt: "x")[$x$]-intercepts.


      
      #notebox("Caution", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Many people think that the inequality signs in the solution should point in the same direction as the sign in the original problem, and hence would incorrectly write the solution to Example as #math.equation(block: false, alt: "x less than or equal to minus 2")[$x ≤ − 2$] or #math.equation(block: false, alt: "x less than or equal to 8")[$x ≤ 8$]. However, you can see from the graph that this is incorrect. Remember that the graph of a quadratic equation is a parabola, not a straight line!
]


      
        Solve #math.equation(block: false, alt: "x squared less than 20")[$x^(2) < 20$].


        Answer: \_\_\_\_\_


        You may use inequality symbols or enter your answer using interval notation. When using inequality symbols, enter "\<=" for #math.equation(block: false, alt: "less than or equal to")[$≤$], and enter "\>=" for #math.equation(block: false, alt: "greater than or equal to")[$≥$]. When using interval notation, use "inf" for #math.equation(block: false, alt: "∞")[$upright(∞)$] and use "U" (an upper case letter u) for the union symbol #math.equation(block: false, alt: "union")[$∪$].


        
          + Write the inequality in standard form.
+ Find the #math.equation(block: false, alt: "x")[$x$]-intercepts of the corresponding graph. Use extraction of roots.
+ Make a rough sketch of the graph.
+ Decide which intervals on the #math.equation(block: false, alt: "x")[$x$]-axis give the correct sign for #math.equation(block: false, alt: "y")[$y$].


        
        #solutionbox[
#math.equation(block: true, alt: "minus the square root of 20 less than x less than the square root of 20")[$− sqrt(20) < x < sqrt(20)$]
]


      
      
        Solve #math.equation(block: false, alt: "x squared less than 20")[$x^(2) < 20$].


        + Write the inequality in standard form.
+ Find the #math.equation(block: false, alt: "x")[$x$]-intercepts of the corresponding graph. Use extraction of roots.
+ Make a rough sketch of the graph.
+ Decide which intervals on the #math.equation(block: false, alt: "x")[$x$]-axis give the correct sign for #math.equation(block: false, alt: "y")[$y$].


        #solutionbox[
#math.equation(block: true, alt: "minus the square root of 20 less than x less than the square root of 20")[$− sqrt(20) < x < sqrt(20)$]
]


      
      If we cannot find the #math.equation(block: false, alt: "x")[$x$]-intercepts of the graph by factoring or extraction of roots, we can use the quadratic formula.


      #examplebox("Example 5")[][
TrailGear, Inc. manufactures camping equipment. The company finds that the profit from producing and selling #math.equation(block: false, alt: "x")[$x$] alpine parkas per month is given, in dollars, by


          #math.equation(block: true, alt: "P equals minus 0.8 x squared plus 320 x minus 25 , 200")[$P = − 0.8 x^(2) + 320 x − 25 , 200$]
          How many parkas should the company produce and sell each month if it must keep the profits above \$#math.equation(block: false, alt: "2000")[$2000$]?


        
        #solutionbox[
+ We would like to solve the inequality
					#math.equation(block: true, alt: "minus 0.8 x squared plus 320 x minus 25 , 200 greater than 2000")[$− 0.8 x^(2) + 320 x − 25 , 200 > 2000$]
			or, subtracting #math.equation(block: false, alt: "2000")[$2000$] from both sides,
					#math.equation(block: true, alt: "minus 0.8 x squared plus 320 x minus 27 , 200 greater than 0")[$− 0.8 x^(2) + 320 x − 27 , 200 > 0$]
+ Consider the function
					#math.equation(block: true, alt: "y equals minus 0.8 x squared plus 320 x minus 27 , 200")[$y = − 0.8 x^(2) + 320 x − 27 , 200$]
			We locate the #math.equation(block: false, alt: "x")[$x$]-intercepts of the graph by setting #math.equation(block: false, alt: "y equals 0")[$y = 0$] and solving for #math.equation(block: false, alt: "x")[$x$]. We will use the quadratic formula to solve the equation
					#math.equation(block: true, alt: "minus 0.8 x squared plus 320 x minus 27 , 200 equals 0")[$− 0.8 x^(2) + 320 x − 27 , 200 = 0$]
			so #math.equation(block: false, alt: "a equals minus 0.8")[$a = − 0.8$], #math.equation(block: false, alt: "b equals 320")[$b = 320$], and #math.equation(block: false, alt: "c equals minus 27 , 200")[$c = − 27 , 200$].
					#math.equation(block: true, alt: "x, equals the fraction minus open parenthesis 320 close parenthesis plus or minus the square root of open parenthesis 320 close parenthesis squared minus 4 open parenthesis minus 0.8 close parenthesis open parenthesis minus 27 , 200 close parenthesis over 2 open parenthesis minus 0.8 close parenthesis; equals the fraction minus 320 plus or minus the square root of 102 , 400 minus 87 , 040 over minus 1.6; equals the fraction minus 320 plus or minus the square root of 15 , 360 over minus 1.6")[$x & = frac(− ( 320 ) ± sqrt(( 320 )^(2) − 4 ( − 0.8 ) ( − 27 "," 200 )), 2 ( − 0.8 )) \  & = frac(− 320 ± sqrt(102 "," 400 − 87 "," 040), − 1.6) \  & = frac(− 320 ± sqrt(15 "," 360), − 1.6)$]
			To two decimal places, the solutions to the equation are #math.equation(block: false, alt: "122.54")[$122.54$] and #math.equation(block: false, alt: "277.46")[$277.46$].
+ The graph of the function is a parabola that opens downward, because the coefficient of #math.equation(block: false, alt: "x squared")[$x^(2)$] is negative.#figure(figph[graph of parabola opening down], alt: "graph of parabola opening down", caption: none)
+ The graph lies above the #math.equation(block: false, alt: "x")[$x$]-axis, and hence #math.equation(block: false, alt: "y greater than 0")[$y > 0$], for #math.equation(block: false, alt: "x")[$x$]-values between the two #math.equation(block: false, alt: "x")[$x$]-intercepts, that is, for #math.equation(block: false, alt: "122.54 less than x less than 277.46")[$122.54 < x < 277.46$]. Because we cannot produce a fraction of a parka, we restrict the interval to the closest whole number #math.equation(block: false, alt: "x")[$x$]-values included, namely #math.equation(block: false, alt: "123")[$123$] and #math.equation(block: false, alt: "277")[$277$]. Thus, TrailGear can produce as few as #math.equation(block: false, alt: "123")[$123$] parkas or as many as #math.equation(block: false, alt: "277")[$277$] parkas per month to keep its profit above \$#math.equation(block: false, alt: "2000")[$2000$].
]
]


      
        What does the notation #math.equation(block: false, alt: "open bracket minus 3 , 3 close bracket")[$[ − 3 , 3 ]$] mean?


        \_\_\_\_\_


        #solutionbox[
All real numbers between #math.equation(block: false, alt: "minus 3")[$− 3$] and #math.equation(block: false, alt: "3")[$3$], including the endpoints.
]


      
      
        What does the notation #math.equation(block: false, alt: "open bracket minus 3 , 3 close bracket")[$[ − 3 , 3 ]$] mean?


        + The point with #math.equation(block: false, alt: "x")[$x$]-coordinate 3 and #math.equation(block: false, alt: "y")[$y$]-coordinate 3.
+ #math.equation(block: false, alt: "x equals 3")[$x = 3$] or #math.equation(block: false, alt: "x equals minus 3")[$x = − 3$]
+ All real numbers between #math.equation(block: false, alt: "minus 3")[$− 3$] and #math.equation(block: false, alt: "3")[$3$], excluding the endpoints.
+ All real numbers between #math.equation(block: false, alt: "minus 3")[$− 3$] and #math.equation(block: false, alt: "3")[$3$], including the endpoints.


      
      
        Solve the inequality #math.equation(block: false, alt: "10 minus 8 x plus x squared greater than 4")[$10 − 8 x + x^(2) > 4$].


        Answer: \_\_\_\_\_


        You may use inequality symbols or enter your answer using interval notation. When using inequality symbols, enter "\<=" for #math.equation(block: false, alt: "less than or equal to")[$≤$], and enter "\>=" for #math.equation(block: false, alt: "greater than or equal to")[$≥$]. When using interval notation, use "inf" for #math.equation(block: false, alt: "∞")[$upright(∞)$] and use "U" (an upper case letter u) for the union symbol #math.equation(block: false, alt: "union")[$∪$].


        
          + Write the inequality in standard form.
+ Find the #math.equation(block: false, alt: "x")[$x$]-intercepts of the corresponding graph. Use extraction of roots.
+ Make a rough sketch of the graph.
+ Decide which intervals on the #math.equation(block: false, alt: "x")[$x$]-axis give the correct sign for #math.equation(block: false, alt: "y")[$y$].


        
        #solutionbox[
#math.equation(block: true, alt: "open parenthesis minus ∞ , 4 minus the square root of 10 close bracket union open bracket 4 plus the square root of 10 , ∞ close parenthesis")[$( − upright(∞) , 4 − sqrt(10) ] ∪ [ 4 + sqrt(10) , upright(∞) )$]
]


      
      
        Solve the inequality #math.equation(block: false, alt: "10 minus 8 x plus x squared greater than 4")[$10 − 8 x + x^(2) > 4$].


        + Write the inequality in standard form.
+ Find the #math.equation(block: false, alt: "x")[$x$]-intercepts of the corresponding graph. Use extraction of roots.
+ Make a rough sketch of the graph.
+ Decide which intervals on the #math.equation(block: false, alt: "x")[$x$]-axis give the correct sign for #math.equation(block: false, alt: "y")[$y$].


        #solutionbox[
#math.equation(block: true, alt: "open parenthesis minus ∞ , 4 minus the square root of 10 close bracket union open bracket 4 plus the square root of 10 , ∞ close parenthesis")[$( − upright(∞) , 4 − sqrt(10) ] ∪ [ 4 + sqrt(10) , upright(∞) )$]
]


      
      
        Explain how to use a graph to solve #math.equation(block: false, alt: "x squared plus 4 x minus 77 less than or equal to 0")[$" " x^(2) + 4 x − 77 ≤ 0$].


        \_\_\_\_\_


      
      
        Explain how to use a graph to solve #math.equation(block: false, alt: "x squared plus 4 x minus 77 less than or equal to 0")[$" " x^(2) + 4 x − 77 ≤ 0$].


      
    
    
      
=== Section Summary


      
        
==== Vocabulary


        Look up the definitions of new terms in the Glossary.


        - Compound inequality
- Interval notation


      
      
        
==== CONCEPTS


        + We can use a graphical technique to solve quadratic inequalities.
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[To Solve a Quadratic Inequality Algebraically:]

+ Write the inequality in standard form: One side is #math.equation(block: false, alt: "0")[$0$], and the other has the form #math.equation(block: false, alt: "a x squared plus b x plus c")[$a x^(2) + b x + c$].
+ Find the #math.equation(block: false, alt: "x")[$x$]-intercepts of the graph of #math.equation(block: false, alt: "y equals a x squared plus b x plus c")[$y = a x^(2) + b x + c$] by setting #math.equation(block: false, alt: "y equals 0")[$y = 0$] and solving for #math.equation(block: false, alt: "x")[$x$].
+ Make a rough sketch of the graph, using the sign of #math.equation(block: false, alt: "a")[$a$] to determine whether the parabola opens upward or downward.
+ Decide which intervals on the #math.equation(block: false, alt: "x")[$x$]-axis give the correct sign for #math.equation(block: false, alt: "y")[$y$].
]


      
      
        
==== STUDY QUESTIONS


        + If #math.equation(block: false, alt: "a x squared plus b x greater than c")[$a x^(2) + b x > c$] for a particular value of #math.equation(block: false, alt: "x")[$x$], what can you say about the graph of #math.equation(block: false, alt: "y equals a x squared plus b x minus c")[$y = a x^(2) + b x − c$] at that #math.equation(block: false, alt: "x")[$x$]-value?
+ What are the only #math.equation(block: false, alt: "x")[$x$]-values at which the graph of #math.equation(block: false, alt: "y equals a x squared plus b x plus c")[$y = a x^(2) + b x + c$] can change sign?
+ Explain the difference between an open interval and a closed interval.
+ Explain what is wrong with the following "solution" to a quadratic inequality: #math.equation(block: false, alt: "2 greater than x greater than 8")[$2 > x > 8$].
+ The parabola #math.equation(block: false, alt: "y equals x squared plus b x plus c")[$y = x^(2) + b x + c$] has #math.equation(block: false, alt: "x")[$x$]-intercepts at #math.equation(block: false, alt: "r sub 1")[$r_(1)$] and #math.equation(block: false, alt: "r sub 2")[$r_(2)$], with #math.equation(block: false, alt: "r sub 1 less than r sub 2")[$r_(1) < r_(2)$]. What are the solutions of the inequality #math.equation(block: false, alt: "x squared plus b x plus c greater than 0")[$x^(2) + b x + c > 0$]?
+ The parabola #math.equation(block: false, alt: "y equals x squared plus b x plus c")[$y = x^(2) + b x + c$] has #math.equation(block: false, alt: "x")[$x$]-intercepts at #math.equation(block: false, alt: "r sub 1")[$r_(1)$] and #math.equation(block: false, alt: "r sub 2")[$r_(2)$], with #math.equation(block: false, alt: "r sub 1 less than r sub 2")[$r_(1) < r_(2)$].  What are the solutions of the inequality #math.equation(block: false, alt: "x squared plus b x plus c less than or equal to 0")[$x^(2) + b x + c ≤ 0$]?
+ The parabola #math.equation(block: false, alt: "y equals minus x squared plus b x plus c")[$y = − x^(2) + b x + c$] has #math.equation(block: false, alt: "x")[$x$]-intercepts at  #math.equation(block: false, alt: "r sub 1")[$r_(1)$] and #math.equation(block: false, alt: "r sub 2")[$r_(2)$], with #math.equation(block: false, alt: "r sub 1 less than r sub 2")[$r_(1) < r_(2)$]. What are the solutions of the inequality #math.equation(block: false, alt: "minus x squared plus b x plus c less than or equal to 0")[$− x^(2) + b x + c ≤ 0$]?
+ The parabola #math.equation(block: false, alt: "y equals minus x squared plus b x plus c")[$y = − x^(2) + b x + c$] has #math.equation(block: false, alt: "x")[$x$]-intercepts at  #math.equation(block: false, alt: "r sub 1")[$r_(1)$] and #math.equation(block: false, alt: "r sub 2")[$r_(2)$], with #math.equation(block: false, alt: "r sub 1 less than r sub 2")[$r_(1) < r_(2)$]. What are the solutions of the inequality #math.equation(block: false, alt: "minus x squared plus b x plus c greater than 0")[$− x^(2) + b x + c > 0$]?


      
      
        
==== SKILLS


        Practice each skill in the Homework problems listed.


        + Solve a quadratic inequality graphically: \#1–30
+ Solve a quadratic inequality algebraically: \#31–50
+ Solve problems involving quadratic inequalities: \#51–60


      
    
    
      
=== Homework 6.5


      
        + Graph the function #math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$] by hand on graph paper.
+ Darken the portion of the #math.equation(block: false, alt: "x")[$x$]-axis for which #math.equation(block: false, alt: "y greater than 9")[$y > 9$].
+ Solve the inequality #math.equation(block: false, alt: "x squared greater than 9")[$x^(2) > 9$]. Explain why #math.equation(block: false, alt: "x greater than 3")[$x > 3$] is incorrect as an answer.


        
          + #figure(figph[parabola for quadratic inequality], alt: "parabola for quadratic inequality", caption: none)
+ See graph
+ #math.equation(block: false, alt: "x greater than 3")[$x > 3$] or #math.equation(block: false, alt: "x less than minus 3")[$x < − 3$]. It omits the solutions #math.equation(block: false, alt: "x less than minus 3")[$x < − 3$].


        
      
      
        + Graph the function #math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$] by hand on graph paper.
+ Darken the portion of the #math.equation(block: false, alt: "x")[$x$]-axis for which #math.equation(block: false, alt: "y less than 1")[$y < 1$].
+ Solve the inequality #math.equation(block: false, alt: "x squared less than 1")[$x^(2) < 1$]. Explain why #math.equation(block: false, alt: "x less than 1")[$x < 1$] is incorrect as an answer.


      
      
        + Graph the function #math.equation(block: false, alt: "y equals x squared minus 2 x minus 3")[$y = x^(2) − 2 x − 3$] by hand on graph paper.
+ Darken the portion of the #math.equation(block: false, alt: "x")[$x$]-axis for which #math.equation(block: false, alt: "y greater than 0")[$y > 0$].
+ Solve the inequality #math.equation(block: false, alt: "x squared minus 2 x minus 3 greater than 0")[$x^(2) − 2 x − 3 > 0$].


        
          + #figure(figph[parabola for quadratic inequality], alt: "parabola for quadratic inequality", caption: none)
+ See graph
+ #math.equation(block: false, alt: "x greater than 3")[$x > 3$] or #math.equation(block: false, alt: "x less than minus 1")[$x < − 1$]


        
      
      
        + Graph the function #math.equation(block: false, alt: "y equals x squared plus 2 x minus 8")[$y = x^(2) + 2 x − 8$] by hand on graph paper.
+ Darken the portion of the #math.equation(block: false, alt: "x")[$x$]-axis for which #math.equation(block: false, alt: "y less than 0")[$y < 0$].
+ Solve the inequality #math.equation(block: false, alt: "x squared plus 2 x minus 8 less than 0")[$x^(2) + 2 x − 8 < 0$].


      
      For problems 5-8, use the graphs provided to estimate the solutions to each equation and inequality.


      
        + #math.equation(block: false, alt: "x squared minus 3 x minus 180 equals 0")[$x^(2) − 3 x − 180 = 0$]
+ #math.equation(block: false, alt: "x squared minus 3 x minus 180 greater than 0")[$x^(2) − 3 x − 180 > 0$]


        #figure(figph[parabola], alt: "parabola", caption: none)


        
          + #math.equation(block: false, alt: "minus 12 , 15")[$− 12 , 15$]
+ #math.equation(block: false, alt: "x less than minus 12")[$x < − 12$] or #math.equation(block: false, alt: "x greater than 15")[$x > 15$]


        
      
      
        + #math.equation(block: false, alt: "175 minus 18 x minus x squared equals 0")[$175 − 18 x − x^(2) = 0$]
+ #math.equation(block: false, alt: "175 minus 18 x minus x squared less than 0")[$175 − 18 x − x^(2) < 0$]


        #figure(figph[parabola], alt: "parabola", caption: none)


      
      
        + #math.equation(block: false, alt: "minus 6 x squared plus 4.8 x minus 0.9 equals 0")[$− 6 x^(2) + 4.8 x − 0.9 = 0$]
+ #math.equation(block: false, alt: "minus 6 x squared plus 4.8 x minus 0.9 greater than or equal to 0")[$− 6 x^(2) + 4.8 x − 0.9 ≥ 0$]


        #figure(figph[parabola], alt: "parabola", caption: none)


        
          + #math.equation(block: false, alt: "0.3 , 0.5")[$0.3 , 0.5$]
+ #math.equation(block: false, alt: "0.3 less than or equal to x less than or equal to 0.5")[$0.3 ≤ x ≤ 0.5$]


        
      
      
        + #math.equation(block: false, alt: "5 x squared plus 7.5 x plus 1.8 equals 0")[$5 x^(2) + 7.5 x + 1.8 = 0$]
+ #math.equation(block: false, alt: "5 x squared plus 7.5 x plus 1.8 less than or equal to 0")[$5 x^(2) + 7.5 x + 1.8 ≤ 0$]


        #figure(figph[parabola], alt: "parabola", caption: none)


      
      For Problems 9-12, graph the parabola in the window


      #math.equation(block: true, alt: "Xmin equals minus 9.4 , Xmax equals 9.4 , Ymin equals minus 25 , Ymax equals 25")[$"Xmin" = − 9.4 , " " " " "Xmax" = 9.4 , " " " " "Ymin" = − 25 , " " " " "Ymax" = 25$]
      Use the graph to solve the inequalities. Write your answers in interval notation.


      
        #math.equation(block: true, alt: "y equals x squared minus 3 x minus 18")[$y = x^(2) − 3 x − 18$]


        (For parts (c) and (d), it may be helpful to graph #math.equation(block: false, alt: "Y sub 2 equals minus 8")[$Y_(2) = − 8$] as well.)


        + #math.equation(block: false, alt: "x squared minus 3 x minus 18 greater than 0")[$x^(2) − 3 x − 18 > 0$]
+ #math.equation(block: false, alt: "x squared minus 3 x minus 18 less than 0")[$x^(2) − 3 x − 18 < 0$]
+ #math.equation(block: false, alt: "x squared minus 3 x minus 18 less than or equal to minus 8")[$x^(2) − 3 x − 18 ≤ − 8$]
+ #math.equation(block: false, alt: "x squared minus 3 x minus 18 greater than or equal to minus 8")[$x^(2) − 3 x − 18 ≥ − 8$]


        
          + #math.equation(block: false, alt: "open parenthesis minus ∞ , minus 3 close parenthesis union open parenthesis 6 , ∞ close parenthesis")[$( − upright(∞) , − 3 ) ∪ ( 6 , upright(∞) )$]
+ #math.equation(block: false, alt: "open parenthesis minus 3 , 6 close parenthesis")[$( − 3 , 6 )$]
+ #math.equation(block: false, alt: "open bracket minus 2 , 5 close bracket")[$[ − 2 , 5 ]$]
+ #math.equation(block: false, alt: "open parenthesis minus ∞ , minus 2 close bracket union open bracket 5 , ∞ close parenthesis")[$( − upright(∞) , − 2 ] ∪ [ 5 , upright(∞) )$]


        
      
      
        #math.equation(block: true, alt: "y equals 16 minus 6 x minus x squared")[$y = 16 − 6 x − x^(2)$]


        (For parts (c) and (d), it may be helpful to graph #math.equation(block: false, alt: "Y sub 2 equals 21")[$Y_(2) = 21$] as well.)


        + #math.equation(block: false, alt: "16 minus 6 x minus x squared greater than or equal to 0")[$16 − 6 x − x^(2) ≥ 0$]
+ #math.equation(block: false, alt: "16 minus 6 x minus x squared less than or equal to 0")[$16 − 6 x − x^(2) ≤ 0$]
+ #math.equation(block: false, alt: "16 minus 6 x minus x squared less than 21")[$16 − 6 x − x^(2) < 21$]
+ #math.equation(block: false, alt: "16 minus 6 x minus x squared greater than 21")[$16 − 6 x − x^(2) > 21$]


      
      
        #math.equation(block: true, alt: "y equals 16 minus x squared")[$y = 16 − x^(2)$]


        (For parts (c) and (d), it may be helpful to graph #math.equation(block: false, alt: "Y sub 2 equals 7")[$Y_(2) = 7$] as well.)


        + #math.equation(block: false, alt: "16 minus x squared greater than 0")[$16 − x^(2) > 0$]
+ #math.equation(block: false, alt: "16 minus x squared less than 0")[$16 − x^(2) < 0$]
+ #math.equation(block: false, alt: "16 minus x squared less than or equal to 7")[$16 − x^(2) ≤ 7$]
+ #math.equation(block: false, alt: "16 minus x squared greater than or equal to 7")[$16 − x^(2) ≥ 7$]


        
          + #math.equation(block: false, alt: "open parenthesis minus 4 , 4 close parenthesis")[$( − 4 , 4 )$]
+ #math.equation(block: false, alt: "open parenthesis minus ∞ , minus 4 close parenthesis union open parenthesis 4 , ∞ close parenthesis")[$( − upright(∞) , − 4 ) ∪ ( 4 , upright(∞) )$]
+ #math.equation(block: false, alt: "open parenthesis minus ∞ , minus 3 close bracket union open bracket 3 , ∞ close parenthesis")[$( − upright(∞) , − 3 ] ∪ [ 3 , upright(∞) )$]
+ #math.equation(block: false, alt: "open bracket minus 3 , 3 close bracket")[$[ − 3 , 3 ]$]


        
      
      
        #math.equation(block: true, alt: "y equals x squared minus 9")[$y = x^(2) − 9$]


        (For parts (c) and (d), it may be helpful to graph #math.equation(block: false, alt: "Y sub 2 equals 16")[$Y_(2) = 16$] as well.)


        + #math.equation(block: false, alt: "x squared minus 9 greater than or equal to 0")[$x^(2) − 9 ≥ 0$]
+ #math.equation(block: false, alt: "x squared minus 9 less than or equal to 0")[$x^(2) − 9 ≤ 0$]
+ #math.equation(block: false, alt: "x squared minus 9 greater than 16")[$x^(2) − 9 > 16$]
+ #math.equation(block: false, alt: "x squared minus 9 less than 16")[$x^(2) − 9 < 16$]


      
      For Problems 13-18, solve the inequality by graphing. Use the following window settings:


      #math.equation(block: true, alt: "Xmin equals minus 9.4 , Xmax equals 9.4 , Ymin equals minus 15 , Ymax equals 15")[$"Xmin" = − 9.4 , " " " " "Xmax" = 9.4 , " " " " "Ymin" = − 15 , " " " " "Ymax" = 15$]
      
        #math.equation(block: true, alt: "open parenthesis x minus 3 close parenthesis open parenthesis x plus 2 close parenthesis greater than 0")[$( x − 3 ) ( x + 2 ) > 0$]


        
          #math.equation(block: true, alt: "open parenthesis minus ∞ , minus 2 close parenthesis union open parenthesis 3 , ∞ close parenthesis")[$( − upright(∞) , − 2 ) ∪ ( 3 , upright(∞) )$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis x plus 1 close parenthesis open parenthesis x minus 5 close parenthesis greater than 0")[$( x + 1 ) ( x − 5 ) > 0$]


      
      
        #math.equation(block: true, alt: "k open parenthesis 4 minus k close parenthesis greater than or equal to 0")[$k ( 4 − k ) ≥ 0$]


        
          #math.equation(block: true, alt: "open bracket 0 , 4 close bracket")[$[ 0 , 4 ]$]


        
      
      
        #math.equation(block: true, alt: "minus m open parenthesis 7 plus m close parenthesis greater than or equal to 0")[$− m ( 7 + m ) ≥ 0$]


      
      
        #math.equation(block: true, alt: "6 plus 5 p minus p squared less than 0")[$6 + 5 p − p^(2) < 0$]


        
          #math.equation(block: true, alt: "open parenthesis ∞ , minus 1 close parenthesis union open parenthesis 6 , ∞ close parenthesis")[$( upright(∞) , − 1 ) ∪ ( 6 , upright(∞) )$]


        
      
      
        #math.equation(block: true, alt: "q squared plus 9 q plus 18 less than 0")[$q^(2) + 9 q + 18 < 0$]


      
      For problems 19-24, solve the inequality by graphing. Use the following window settings:


      #math.equation(block: true, alt: "Xmin equals minus 9.4 , Xmax equals 9.4 , Ymin equals minus 62 , Ymax equals 62")[$"Xmin" = − 9.4 , " " " " "Xmax" = 9.4 , " " " " "Ymin" = − 62 , " " " " "Ymax" = 62$]
      
        #math.equation(block: true, alt: "x squared minus 1.4 x minus 20 less than 9.76")[$x^(2) − 1.4 x − 20 < 9.76$]


        
          #math.equation(block: true, alt: "open parenthesis minus 4.8 , 6.2 close parenthesis")[$( − 4.8 , 6.2 )$]


        
      
      
        #math.equation(block: true, alt: "minus x squared plus 3.2 x plus 20 greater than 6.56")[$− x^(2) + 3.2 x + 20 > 6.56$]


      
      
        #math.equation(block: true, alt: "5 x squared plus 39 x plus 27 greater than or equal to 5.4")[$5 x^(2) + 39 x + 27 ≥ 5.4$]


        
          #math.equation(block: true, alt: "open parenthesis minus ∞ , minus 7.2 close bracket union open bracket 0.6 , ∞ close parenthesis")[$( − upright(∞) , − 7.2 ] ∪ [ 0.6 , upright(∞) )$]


        
      
      
        #math.equation(block: true, alt: "minus 6 x squared minus 36 x minus 20 less than or equal to 25.36")[$− 6 x^(2) − 36 x − 20 ≤ 25.36$]


      
      
        #math.equation(block: true, alt: "minus 8 x squared plus 112 x minus 360 less than 6.08")[$− 8 x^(2) + 112 x − 360 < 6.08$]


        
          #math.equation(block: true, alt: "open parenthesis minus ∞ , 5.2 close parenthesis union open parenthesis 8.8 , ∞ close parenthesis")[$( − upright(∞) , 5.2 ) ∪ ( 8.8 , upright(∞) )$]


        
      
      
        #math.equation(block: true, alt: "10 x squared plus 96 x plus 180 greater than 17.2")[$10 x^(2) + 96 x + 180 > 17.2$]


      
      For problems 25-30, solve the inequality by graphing. Choose a suitable window for each problem. Use the intersect feature to estimate your solutions accurate to one decimal place.


      
        #math.equation(block: true, alt: "x squared greater than 12.2")[$x^(2) > 12.2$]


        
          #math.equation(block: false, alt: "x less than minus 3.5")[$x < − 3.5$] or #math.equation(block: false, alt: "x greater than 3.5")[$x > 3.5$]


        
      
      
        #math.equation(block: true, alt: "x squared less than or equal to 45")[$x^(2) ≤ 45$]


      
      
        #math.equation(block: true, alt: "minus 3 x squared plus 7 x minus 25 less than or equal to 0")[$− 3 x^(2) + 7 x − 25 ≤ 0$]


        
          All #math.equation(block: false, alt: "x")[$x$]


        
      
      
        #math.equation(block: true, alt: "2.4 x squared minus 5.6 x plus 18 less than or equal to 0")[$2.4 x^(2) − 5.6 x + 18 ≤ 0$]


      
      
        #math.equation(block: true, alt: "0.4 x squared minus 54 x less than 620")[$0.4 x^(2) − 54 x < 620$]


        
          #math.equation(block: true, alt: "minus 10.6 less than x less than 145.6")[$− 10.6 < x < 145.6$]


        
      
      
        #math.equation(block: true, alt: "minus 0.05 x squared minus 3 x greater than 76")[$− 0.05 x^(2) − 3 x > 76$]


      
      For Problems 31-50, solve the inequality algebraically.Write your answers in interval notation, rounding to two decimal places if necessary.


      
        #math.equation(block: true, alt: "open parenthesis x plus 3 close parenthesis open parenthesis x minus 4 close parenthesis less than 0")[$( x + 3 ) ( x − 4 ) < 0$]


        
          #math.equation(block: true, alt: "open parenthesis minus 3 , 4 close parenthesis")[$( − 3 , 4 )$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis x plus 2 close parenthesis open parenthesis x plus 5 close parenthesis greater than 0")[$( x + 2 ) ( x + 5 ) > 0$]


      
      
        #math.equation(block: true, alt: "28 minus 3 x minus x squared greater than or equal to 0")[$28 − 3 x − x^(2) ≥ 0$]


        
          #math.equation(block: true, alt: "open bracket minus 7 , 4 close bracket")[$[ − 7 , 4 ]$]


        
      
      
        #math.equation(block: true, alt: "32 plus 4 x minus x squared less than or equal to 0")[$32 + 4 x − x^(2) ≤ 0$]


      
      
        #math.equation(block: true, alt: "2 z squared minus 7 z greater than 4")[$2 z^(2) − 7 z > 4$]


        
          #math.equation(block: true, alt: "open parenthesis minus ∞ , the fraction minus 1 over 2 close parenthesis union open parenthesis 4 , ∞ close parenthesis")[$( − upright(∞) , display(frac(− 1, 2)) ) ∪ ( 4 , upright(∞) )$]


        
      
      
        #math.equation(block: true, alt: "6 h squared plus 13 h less than 15")[$6 h^(2) + 13 h < 15$]


      
      
        #math.equation(block: true, alt: "64 minus t squared greater than 0")[$64 − t^(2) > 0$]


        
          #math.equation(block: true, alt: "open parenthesis minus 8 , 8 close parenthesis")[$( − 8 , 8 )$]


        
      
      
        #math.equation(block: true, alt: "121 minus y squared less than 0")[$121 − y^(2) < 0$]


      
      
        #math.equation(block: true, alt: "v squared less than 5")[$v^(2) < 5$]


        
          #math.equation(block: true, alt: "open parenthesis minus 2.24 , 2.24 close parenthesis")[$( − 2.24 , 2.24 )$]


        
      
      
        #math.equation(block: true, alt: "t squared greater than or equal to 7")[$t^(2) ≥ 7$]


      
      
        #math.equation(block: true, alt: "5 a squared minus 32 a plus 12 greater than or equal to 0")[$5 a^(2) − 32 a + 12 ≥ 0$]


        
          #math.equation(block: true, alt: "open parenthesis minus ∞ , 0.4 close bracket union open bracket 6 , ∞ close parenthesis")[$( − upright(∞) , 0.4 ] ∪ [ 6 , upright(∞) )$]


        
      
      
        #math.equation(block: true, alt: "6 b squared plus 16 b minus 9 less than 0")[$6 b^(2) + 16 b − 9 < 0$]


      
      
        #math.equation(block: true, alt: "4 x squared plus x greater than or equal to minus 2 x squared plus 2")[$4 x^(2) + x ≥ − 2 x^(2) + 2$]


        
          #math.equation(block: true, alt: "open parenthesis minus ∞ , minus 0.67 close parenthesis union open parenthesis 0.5 , ∞ close parenthesis")[$( − upright(∞) , − 0.67 ) ∪ ( 0.5 , upright(∞) )$]


        
      
      
        #math.equation(block: true, alt: "2 x squared plus 8 x less than or equal to minus x squared plus 3")[$2 x^(2) + 8 x ≤ − x^(2) + 3$]


      
      
        #math.equation(block: true, alt: "x squared minus 4 x plus 1 greater than or equal to 0")[$x^(2) − 4 x + 1 ≥ 0$]


        
          #math.equation(block: true, alt: "open parenthesis minus ∞ , 0.27 close bracket union open bracket 3.73 , ∞ close parenthesis")[$( − upright(∞) , 0.27 ] ∪ [ 3.73 , upright(∞) )$]


        
      
      
        #math.equation(block: true, alt: "x squared plus 4 x plus 2 less than or equal to 0")[$x^(2) + 4 x + 2 ≤ 0$]


      
      
        #math.equation(block: true, alt: "minus 3 minus m squared less than 0")[$− 3 − m^(2) < 0$]


        
          All #math.equation(block: false, alt: "m")[$m$]


        
      
      
        #math.equation(block: true, alt: "11 plus n squared less than 0")[$11 + n^(2) < 0$]


      
      
        #math.equation(block: true, alt: "w squared minus w plus 4 less than or equal to 0")[$w^(2) − w + 4 ≤ 0$]


        
          No solution


        
      
      
        #math.equation(block: true, alt: "minus z squared plus z minus 1 less than or equal to 0")[$− z^(2) + z − 1 ≤ 0$]


      
      In Problems 51–58,


      + Solve each problem by writing and solving an inequality.
+ Graph the equation and verify your solution on the graph.


      
        A fireworks rocket is fired from ground level. Its height in feet #math.equation(block: false, alt: "t")[$t$] seconds after launch is given by


        #math.equation(block: true, alt: "h equals 320 t minus 16 t squared")[$h = 320 t − 16 t^(2)$]
        During what time interval is the rocket higher than #math.equation(block: false, alt: "1024")[$1024$] feet?


        
          + #math.equation(block: false, alt: "320 t minus 16 t squared greater than 1024")[$320 t − 16 t^(2) > 1024$]; #math.equation(block: false, alt: "4 less than t less than 16")[$4 < t < 16$]: Between #math.equation(block: false, alt: "4")[$4$] and #math.equation(block: false, alt: "16")[$16$] seconds
+ #figure(figph[parabola for inequality], alt: "parabola for inequality", caption: none)


        
      
      
        A baseball thrown vertically reaches a height, #math.equation(block: false, alt: "h")[$h$], in feet given by


        #math.equation(block: true, alt: "h equals 56 t minus 16 t squared")[$h = 56 t − 16 t^(2)$]
        where #math.equation(block: false, alt: "t")[$t$] is measured in seconds. During what time intervals is the ball between #math.equation(block: false, alt: "40")[$40$] and #math.equation(block: false, alt: "48")[$48$] feet high?


      
      
        The cost, in dollars, of manufacturing #math.equation(block: false, alt: "x")[$x$] pairs of garden shears is given by the function


        #math.equation(block: true, alt: "C equals minus 0.02 x squared plus 14 x plus 1600")[$C = − 0.02 x^(2) + 14 x + 1600$]
        for #math.equation(block: false, alt: "0 less than or equal to x less than or equal to 700")[$0 ≤ x ≤ 700$]. How many pairs of shears can be
produced if the total cost must be kept under \$#math.equation(block: false, alt: "2800")[$2800$]?


        
          + #math.equation(block: false, alt: "minus 0.02 x squared plus 14 x plus 1600 less than 2800")[$− 0.02 x^(2) + 14 x + 1600 < 2800$]; #math.equation(block: false, alt: "open bracket 0 , 100 close parenthesis union open parenthesis 600 , 700 close bracket")[$[ 0 , 100 ) ∪ ( 600 , 700 ]$]: Either less than #math.equation(block: false, alt: "100")[$100$] or more than #math.equation(block: false, alt: "600")[$600$] shears.
+ #figure(figph[parabola for inequality], alt: "parabola for inequality", caption: none)


        
      
      
        The cost, in dollars, of producing #math.equation(block: false, alt: "x")[$x$] cashmere sweaters is given by the function


        #math.equation(block: true, alt: "C equals x squared plus 4 x plus 90")[$C = x^(2) + 4 x + 90$]
        How many sweaters can be produced if the total cost
must be kept under #math.equation(block: false, alt: "$ 1850")[$\$ 1850$]?


      
      
        The Locker Room finds that it sells #math.equation(block: false, alt: "1200 minus 30 p")[$1200 − 30 p$] sweatshirts each month when it charges #math.equation(block: false, alt: "p")[$p$] dollars per sweatshirt. It would like its revenue from sweatshirts to be over \$#math.equation(block: false, alt: "9000")[$9000$] per month. In what interval should it keep the price of a sweatshirt?


        
          + #math.equation(block: false, alt: "p open parenthesis 1200 minus 30 p close parenthesis greater than 9000")[$p ( 1200 − 30 p ) > 9000$]; #math.equation(block: false, alt: "10 less than p less than 30")[$10 < p < 30$]: Between \$#math.equation(block: false, alt: "10")[$10$] and \$#math.equation(block: false, alt: "30")[$30$]
+ #figure(figph[parabola for inequality], alt: "parabola for inequality", caption: none)


        
      
      
        Green Valley Nursery sells #math.equation(block: false, alt: "120 minus 10 p")[$120 − 10 p$] boxes of rose food per month at a price of #math.equation(block: false, alt: "p")[$p$] dollars per box. It would like to keep its monthly revenue from rose food over \$#math.equation(block: false, alt: "350")[$350$]. In what interval should it price a box of rose food?


      
      
        A group of cylindrical storage tanks must be #math.equation(block: false, alt: "20")[$20$] feet tall. If the volume of each tank must be between #math.equation(block: false, alt: "500 π")[$500 π$] and #math.equation(block: false, alt: "2880 π")[$2880 π$] cubic feet, what are the possible values for the
radius of a tank?


        
          + #math.equation(block: false, alt: "500 π less than 20 π r squared less than 2880 π")[$500 π < 20 π r^(2) < 2880 π$]; #math.equation(block: false, alt: "5 less than r less than 12")[$5 < r < 12$]; The radius must be between #math.equation(block: false, alt: "5")[$5$] and #math.equation(block: false, alt: "12")[$12$] ft.
+ #figure(figph[parabola for inequality], alt: "parabola for inequality", caption: none)


        
      
      
        The volume of a cylindrical can should be between #math.equation(block: false, alt: "21.2")[$21.2$] and #math.equation(block: false, alt: "21.6")[$21.6$] cubic inches. If the height of the can is #math.equation(block: false, alt: "5")[$5$] inches, what values for the radius (to the nearest hundredth of an inch) will produce an acceptable can?


      
      
        A travel agency offers a group rate of \$#math.equation(block: false, alt: "600")[$600$] per person for a weekend in Lake Tahoe if #math.equation(block: false, alt: "20")[$20$] people sign up. For each additional person who signs up, the price for all participants is reduced by \$#math.equation(block: false, alt: "10")[$10$] per person.


        + Write algebraic expressions for the size of the group and the price per person if #math.equation(block: false, alt: "x")[$x$] additional people sign up.
+ Write a formula for the travel agency's total income as a function of #math.equation(block: false, alt: "x")[$x$].
+ What is the maximum income the travel agency can earn on the Lake Tahoe weekend? How many people should the agency enroll to achieve this income?
+ How many people must sign up in order for the agency to bring in at least \$#math.equation(block: false, alt: "15 , 750")[$15 , 750$]?
+ Graph the income function and use the graph to verify your answers to parts (c) and (d).


        
          + Size of group: #math.equation(block: false, alt: "20 plus x")[$20 + x$]; Price per person: #math.equation(block: false, alt: "600 minus 10 x")[$600 − 10 x$]
+ #math.equation(block: false, alt: "I equals open parenthesis 20 plus x close parenthesis open parenthesis 600 minus 10 x close parenthesis")[$I = ( 20 + x ) ( 600 − 10 x )$]
+ \$#math.equation(block: false, alt: "16 , 000")[$16 , 000$]; #math.equation(block: false, alt: "20")[$" " 20$]
+ Between 5 and 35
+ #figure(figph[parabola for inequality], alt: "parabola for inequality", caption: none)


        
      
      
        A farmer inherits an apple orchard on which #math.equation(block: false, alt: "60")[$60$] trees are planted per acre. Each tree yields 12 bushels of apples. Experimentation has shown that for each tree removed per acre, the yield per tree increases by #math.equation(block: false, alt: "the fraction 1 over 2")[$display(frac(1, 2))$] bushel.


        + Write algebraic expressions for the number of trees per acre and for the yield per tree if #math.equation(block: false, alt: "x")[$x$] trees per acre are removed.
+ Write a formula for the total yield per acre as a function of #math.equation(block: false, alt: "x")[$x$].
+ What is the maximum yield per acre that can be achieved by removing trees? How many trees per acre should be removed to achieve this yield?
+ How many trees should be removed per acre in order to harvest at least 850 bushels per acre?
+ Graph the yield function and use the graph to verify your answers to parts (c) and (d).