6.5 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.

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 $800$ 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

$h = f (t) = 256 t - 16 t^{2}$

When should the Starburst explode in order to satisfy the safety regulation?

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

When is the rocket's height greater than 800 feet, or, in mathematical terms, for what values of $t$ is $h greater than 800$ ? The answer to this question is the solution of the inequality

$256 t minus 16 t squared greater than 800$

Points on the graph with $h greater than 800$ are shown in color, and the $t$ -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 $4.25$ seconds to $11.75$ seconds after launch. The solution of the inequality is the set of all $t$ -values greater than $4.25$ but less than $11.75$ .

What are the solutions of the inequality $x squared plus 1 less than 0$ ?

_____

No solution

What are the solutions of the inequality $x squared plus 1 less than 0$ ?

$open parenthesis minus 1, 1 close parenthesis$
$(- \infty, - 1) \cup (1, \infty)$
All real numbers
No solution

Technology

Solving an Inequality With a Graphing Calculator

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

$\begin{aligned} Y_{1} & = 256 X - 16 X^{2} \\ Y_{2} & = 800 \end{aligned}$

on the same screen. Use WINDOW settings to match the graph in Example.

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

Graph the function $y = x^{2} - 2 x - 9$ in the window
$\begin{aligned} Xmin & = - 0.4 & Xmax = 9.4 \\ Ymin & = - 10 & Ymax = 10 \end{aligned}$
Use the graph to solve the inequality $x^{2} - 2 x - 9 \geq 6$ .
Solution: _____

You may use inequality symbols or enter your answer using interval notation. When using inequality symbols, enter "<=" for $\leq$ , and enter ">=" for $\geq$ . When using interval notation, use "inf" for $∞$ and use "U" (an upper case letter u) for the union symbol $\cup$ .

A graph is below.
$x less than or equal to minus 3$ or $x greater than or equal to 5$

A graph for part (a):

Graph the function $y = x^{2} - 2 x - 9$ in the window
$\begin{aligned} Xmin & = - 0.4 & Xmax = 9.4 \\ Ymin & = - 10 & Ymax = 10 \end{aligned}$
Use the graph to solve the inequality $x^{2} - 2 x - 9 \geq 6$ .

$x less than or equal to minus 3$ or $x greater than or equal to 5$

In Example, we solved the inequality $256 t minus 16 t squared greater than 800$ by comparing points on the graph of $h = 256 - 16 t^{2}$ with points on the line $h equals 800$ . If one side of an inequality is zero, we can compare points on the graph with the line $y equals 0$ , which is the $x$ -axis.

Example 2

Consider the graph of $y = x^{2} - 4$ . Find the solutions of the following equations and inequalities.

$x^{2} - 4 = 0$
$x squared minus 4 less than 0$
$x squared minus 4 greater than 0$

Look at the graph of $y = x^{2} - 4$ shown below. When we substitute a value of $x$ into the expression $x squared minus 4$ , 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.)

$x$	$minus 3$	$minus 2$	$minus 1$	$0$	$1$	$2$	$3$
$y$

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

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

Use a graph of $y = 6 - x^{2}$ to solve the inequalities.

$6 minus x squared greater than 0$
Solution: _____
$6 minus x squared less than or equal to 0$
Solution: _____

$- \sqrt{6} < x < \sqrt{6}$
$x \leq - \sqrt{6}$ or $x greater than or equal to the square root of 6$

Use a graph of $y = 6 - x^{2}$ to solve the inequalities.

$6 minus x squared greater than 0$
$6 minus x squared less than or equal to 0$

$- \sqrt{6} < x < \sqrt{6}$
$x \leq - \sqrt{6}$ or $x greater than or equal to the square root of 6$

Which points on the parabola $y = a x^{2} + b x + c$ do we need to know to solve the inequality $a x^{2} + b x + c > 0$ ?

_____

The $x$ -intercepts

Which points on the parabola $y = a x^{2} + b x + c$ do we need to know to solve the inequality $a x^{2} + b x + c > 0$ ?

The $x$ -intercepts
The $y$ -intercept
The vertex
All of these

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

Example 3

Use a graph to solve $x^{2} - 2 x - 3 \leq 12$

We first write the inequality with zero on one side:

$x^{2} - 2 x - 15 \leq 0$

. We would like to find points on the graph of $y = x^{2} - 2 x - 15$ that have $y$ -coordinates less than or equal to zero. A graph of the equation is shown below.

You can check that the $x$ -intercepts of the graph are $minus 3$ and $5$ . The points shown in red on the graph lie below the $x$ -axis and have $y less than or equal to 0$ , so the $x$ -coordinates of these points are the solutions of the inequality. All of these points have $x$ -coordinates between $minus 3$ and $5$ . Thus, the solution is $minus 3 less than or equal to x less than or equal to 5$ , or in interval notation, $[ minus 3, 5 ]$ .

Follow the steps below to solve the inequality $36 + 6 x - x^{2} \leq 20$ .

Rewrite the inequality so that the right side is zero.
_____ $less than or equal to 0$
Graph the equation $y = 16 + 6 x - x^{2}$ .
Locate the points on the graph with $y$ -coordinate less than zero, and mark the $x$ -coordinates of the points on the $x$ -axis.
Write the solution with interval notation.
Solution: _____

When using interval notation, use "inf" for $∞$ and use "U" (an upper case letter u) for the union symbol $\cup$ .

$16 + 6 x - x^{2} \leq 0$
A graph is below.
See graph
$(- \infty, - 2] \cup [8, \infty)$

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

Follow the steps below to solve the inequality $36 + 6 x - x^{2} \leq 20$ .

Rewrite the inequality so that the right side is zero.
Graph the equation $y = 16 + 6 x - x^{2}$ .
Locate the points on the graph with $y$ -coordinate less than zero, and mark the $x$ -coordinates of the points on the $x$ -axis.
Write the solution with interval notation.

$16 + 6 x - x^{2} \leq 0$
See graph
$(- \infty, - 2] \cup [8, \infty)$

Explain why you cannot write the solutions to $x squared minus 4 greater than 0$ as a single inequality.

_____

Explain why you cannot write the solutions to $x squared minus 4 greater than 0$ 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

$\begin{aligned} a x^{2} + b x + c & < 0, & a x^{2} + b x + c & > 0 \\ a x^{2} + b x + c & \leq 0, & a x^{2} + b x + c & \geq 0 \end{aligned}$

All we really need to know is whether the corresponding parabola $y = a x^{2} + b x + c$ opens upward or downward. Consider the parabolas shown below.

The parabola in figure (a) opens upward. It crosses the $x$ -axis at two points, $x equals r sub 1$ and $x equals r sub 2$ . At these points, $y equals 0$ .

The graph lies below the $x$ -axis between $r sub 1$ and $r sub 2$ , so the solutions to the inequality $y less than 0$ lie between $r sub 1$ and $r sub 2$ .
The graph lies above the $x$ -axis for $x$ -values less than $r sub 1$ or greater than $r sub 2$ , so the solutions to the inequality $y greater than 0$ are $x less than r sub 1$ or $x greater than r sub 2$ .

If the parabola opens downward, as in figure (b), the situation is reversed. The solutions to the inequality $y greater than 0$ lie between the $x$ -intercepts, and the solutions to $y less than 0$ lie outside the $x$ -intercepts.

From the graphs, we see that the $x$ -intercepts are the boundary points between the portions of the graph with positive $y$ -coordinates and the portions with negative $y$ -coordinates. To solve a quadratic inequality, we need only locate the $x$ -intercepts of the corresponding graph and then decide which intervals of the $x$ -axis produce the correct sign for $y$ .

Example 4

Solve the inequality $36 + 6 x - x^{2} \leq 20$ algebraically.

We subtract $20$ from both sides of the inequality so that we have $0$ on the right side.
$16 + 6 x - x^{2} \leq 0$
Consider the equation $y = 16 + 6 x - x^{2}$ . To locate the $x$ -intercepts, we set $y equals 0$ and solve for $x$ .
$\begin{aligned} 16 + 6 x - x^{2} & = 0 & Multiply each term by - 1. \\ x^{2} - 6 x - 16 & = 0 & Factor the left side. \\ (x - 8) (x + 2) & = 0 & Apply the zero-factor principle. \\ x - 8 = 0 or x + 2 & = 0 \\ x = 8 or x & = - 2 \end{aligned}$
The $x$ -intercepts are $x = - 2$ and $x equals 8$ .
Make a rough sketch of the graph of $y = 16 + 6 x - x^{2}$ , as shown below. Because $a = - 1 < 0$ , the graph is a parabola that opens downward.
We are interested in points on the graph for which $y less than or equal to 0$ . The points with negative $y$ -coordinates (that is, points below the $x$ -axis) lie outside the $x$ -intercepts of the graph, so the solution of the inequality is $x less than or equal to minus 2$ or $x greater than or equal to 8$ . Or, using interval notation, the solution is $(- \infty, - 2] \cup [8, \infty)$ .

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

_____

To decide whether the solutions lie between the $x$ -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 $>$ or $<$ in the solution.
To decide whether the solutions lie between the $x$ -intercepts or outside them.
To decide whether the solutions are positive or negative.
To help us find the $x$ -intercepts.

Solve $x squared less than 20$ .

Answer: _____

Write the inequality in standard form.
Find the $x$ -intercepts of the corresponding graph. Use extraction of roots.
Make a rough sketch of the graph.
Decide which intervals on the $x$ -axis give the correct sign for $y$ .

$- \sqrt{20} < x < \sqrt{20}$

Solve $x squared less than 20$ .

Write the inequality in standard form.
Find the $x$ -intercepts of the corresponding graph. Use extraction of roots.
Make a rough sketch of the graph.
Decide which intervals on the $x$ -axis give the correct sign for $y$ .

$- \sqrt{20} < x < \sqrt{20}$

If we cannot find the $x$ -intercepts of the graph by factoring or extraction of roots, we can use the quadratic formula.

Example 5

TrailGear, Inc. manufactures camping equipment. The company finds that the profit from producing and selling $x$ alpine parkas per month is given, in dollars, by

$P = - 0.8 x^{2} + 320 x - 25, 200$

How many parkas should the company produce and sell each month if it must keep the profits above $ $2000$ ?

We would like to solve the inequality
$- 0.8 x^{2} + 320 x - 25, 200 > 2000$
or, subtracting $2000$ from both sides,
$- 0.8 x^{2} + 320 x - 27, 200 > 0$
Consider the function
$y = - 0.8 x^{2} + 320 x - 27, 200$
We locate the $x$ -intercepts of the graph by setting $y equals 0$ and solving for $x$ . We will use the quadratic formula to solve the equation
$- 0.8 x^{2} + 320 x - 27, 200 = 0$
so $a = - 0.8$ , $b equals 320$ , and $c = - 27, 200$ .
$\begin{aligned} x & = \frac{- (320) \pm \sqrt{(320)^{2} - 4 (- 0.8) (- 27, 200)}}{2 (- 0.8)} \\ = \frac{- 320 \pm \sqrt{102, 400 - 87, 040}}{- 1.6} \\ = \frac{- 320 \pm \sqrt{15, 360}}{- 1.6} \end{aligned}$
To two decimal places, the solutions to the equation are $122.54$ and $277.46$ .
The graph of the function is a parabola that opens downward, because the coefficient of $x^{2}$ is negative.
The graph lies above the $x$ -axis, and hence $y greater than 0$ , for $x$ -values between the two $x$ -intercepts, that is, for $122.54 less than x less than 277.46$ . Because we cannot produce a fraction of a parka, we restrict the interval to the closest whole number $x$ -values included, namely $123$ and $277$ . Thus, TrailGear can produce as few as $123$ parkas or as many as $277$ parkas per month to keep its profit above $ $2000$ .

What does the notation $[ minus 3, 3 ]$ mean?

_____

All real numbers between $minus 3$ and $3$ , including the endpoints.

What does the notation $[ minus 3, 3 ]$ mean?

The point with $x$ -coordinate 3 and $y$ -coordinate 3.
$x equals 3$ or $x = - 3$
All real numbers between $minus 3$ and $3$ , excluding the endpoints.
All real numbers between $minus 3$ and $3$ , including the endpoints.

Solve the inequality $10 - 8 x + x^{2} > 4$ .

Answer: _____

Write the inequality in standard form.
Find the $x$ -intercepts of the corresponding graph. Use extraction of roots.
Make a rough sketch of the graph.
Decide which intervals on the $x$ -axis give the correct sign for $y$ .

$(- \infty, 4 - \sqrt{10}] \cup [4 + \sqrt{10}, \infty)$

Solve the inequality $10 - 8 x + x^{2} > 4$ .

Write the inequality in standard form.
Find the $x$ -intercepts of the corresponding graph. Use extraction of roots.
Make a rough sketch of the graph.
Decide which intervals on the $x$ -axis give the correct sign for $y$ .

$(- \infty, 4 - \sqrt{10}] \cup [4 + \sqrt{10}, \infty)$

Explain how to use a graph to solve $x^{2} + 4 x - 77 \leq 0$ .

_____

Explain how to use a graph to solve $x^{2} + 4 x - 77 \leq 0$ .

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.
Note
To Solve a Quadratic Inequality Algebraically:
1. Write the inequality in standard form: One side is $0$ , and the other has the form $a x^{2} + b x + c$ .
2. Find the $x$ -intercepts of the graph of $y = a x^{2} + b x + c$ by setting $y equals 0$ and solving for $x$ .
3. Make a rough sketch of the graph, using the sign of $a$ to determine whether the parabola opens upward or downward.
4. Decide which intervals on the $x$ -axis give the correct sign for $y$ .

STUDY QUESTIONS

If $a x^{2} + b x > c$ for a particular value of $x$ , what can you say about the graph of $y = a x^{2} + b x - c$ at that $x$ -value?
What are the only $x$ -values at which the graph of $y = a x^{2} + b x + c$ 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: $2 greater than x greater than 8$ .
The parabola $y = x^{2} + b x + c$ has $x$ -intercepts at $r sub 1$ and $r sub 2$ , with $r sub 1 less than r sub 2$ . What are the solutions of the inequality $x^{2} + b x + c > 0$ ?
The parabola $y = x^{2} + b x + c$ has $x$ -intercepts at $r sub 1$ and $r sub 2$ , with $r sub 1 less than r sub 2$ . What are the solutions of the inequality $x^{2} + b x + c \leq 0$ ?
The parabola $y = - x^{2} + b x + c$ has $x$ -intercepts at $r sub 1$ and $r sub 2$ , with $r sub 1 less than r sub 2$ . What are the solutions of the inequality $- x^{2} + b x + c \leq 0$ ?
The parabola $y = - x^{2} + b x + c$ has $x$ -intercepts at $r sub 1$ and $r sub 2$ , with $r sub 1 less than r sub 2$ . What are the solutions of the inequality $- x^{2} + b x + c > 0$ ?

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 $y = x^{2}$ by hand on graph paper.
Darken the portion of the $x$ -axis for which $y greater than 9$ .
Solve the inequality $x squared greater than 9$ . Explain why $x greater than 3$ is incorrect as an answer.

See graph
$x greater than 3$ or $x less than minus 3$ . It omits the solutions $x less than minus 3$ .

Graph the function $y = x^{2}$ by hand on graph paper.
Darken the portion of the $x$ -axis for which $y less than 1$ .
Solve the inequality $x squared less than 1$ . Explain why $x less than 1$ is incorrect as an answer.

Graph the function $y = x^{2} - 2 x - 3$ by hand on graph paper.
Darken the portion of the $x$ -axis for which $y greater than 0$ .
Solve the inequality $x^{2} - 2 x - 3 > 0$ .

See graph
$x greater than 3$ or $x less than minus 1$

Graph the function $y = x^{2} + 2 x - 8$ by hand on graph paper.
Darken the portion of the $x$ -axis for which $y less than 0$ .
Solve the inequality $x^{2} + 2 x - 8 < 0$ .

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

$x^{2} - 3 x - 180 = 0$
$x^{2} - 3 x - 180 > 0$

$minus 12, 15$
$x less than minus 12$ or $x greater than 15$

$175 - 18 x - x^{2} = 0$
$175 - 18 x - x^{2} < 0$

$- 6 x^{2} + 4.8 x - 0.9 = 0$
$- 6 x^{2} + 4.8 x - 0.9 \geq 0$

$0.3, 0.5$
$0.3 less than or equal to x less than or equal to 0.5$

$5 x^{2} + 7.5 x + 1.8 = 0$
$5 x^{2} + 7.5 x + 1.8 \leq 0$

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

$Xmin = - 9.4, Xmax = 9.4, Ymin = - 25, Ymax = 25$

Use the graph to solve the inequalities. Write your answers in interval notation.

$y = x^{2} - 3 x - 18$

(For parts (c) and (d), it may be helpful to graph $Y_{2} = - 8$ as well.)

$x^{2} - 3 x - 18 > 0$
$x^{2} - 3 x - 18 < 0$
$x^{2} - 3 x - 18 \leq - 8$
$x^{2} - 3 x - 18 \geq - 8$

$(- \infty, - 3) \cup (6, \infty)$
$open parenthesis minus 3, 6 close parenthesis$
$[ minus 2, 5 ]$
$(- \infty, - 2] \cup [5, \infty)$

$y = 16 - 6 x - x^{2}$

(For parts (c) and (d), it may be helpful to graph $Y sub 2 equals 21$ as well.)

$16 - 6 x - x^{2} \geq 0$
$16 - 6 x - x^{2} \leq 0$
$16 - 6 x - x^{2} < 21$
$16 - 6 x - x^{2} > 21$

$y = 16 - x^{2}$

(For parts (c) and (d), it may be helpful to graph $Y sub 2 equals 7$ as well.)

$16 minus x squared greater than 0$
$16 minus x squared less than 0$
$16 minus x squared less than or equal to 7$
$16 minus x squared greater than or equal to 7$

$open parenthesis minus 4, 4 close parenthesis$
$(- \infty, - 4) \cup (4, \infty)$
$(- \infty, - 3] \cup [3, \infty)$
$[ minus 3, 3 ]$

$y = x^{2} - 9$

(For parts (c) and (d), it may be helpful to graph $Y sub 2 equals 16$ as well.)

$x squared minus 9 greater than or equal to 0$
$x squared minus 9 less than or equal to 0$
$x squared minus 9 greater than 16$
$x squared minus 9 less than 16$

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

$Xmin = - 9.4, Xmax = 9.4, Ymin = - 15, Ymax = 15$

$(x - 3) (x + 2) > 0$

$(- \infty, - 2) \cup (3, \infty)$

$(x + 1) (x - 5) > 0$

$k open parenthesis 4 minus k close parenthesis greater than or equal to 0$

$[ 0, 4 ]$

$- m (7 + m) \geq 0$

$6 + 5 p - p^{2} < 0$

$open parenthesis ∞, minus 1 close parenthesis union open parenthesis 6, ∞ close parenthesis$

$q^{2} + 9 q + 18 < 0$

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

$Xmin = - 9.4, Xmax = 9.4, Ymin = - 62, Ymax = 62$

$x^{2} - 1.4 x - 20 < 9.76$

$open parenthesis minus 4.8, 6.2 close parenthesis$

$- x^{2} + 3.2 x + 20 > 6.56$

$5 x^{2} + 39 x + 27 \geq 5.4$

$(- \infty, - 7.2] \cup [0.6, \infty)$

$- 6 x^{2} - 36 x - 20 \leq 25.36$

$- 8 x^{2} + 112 x - 360 < 6.08$

$open parenthesis minus ∞, 5.2 close parenthesis union open parenthesis 8.8, ∞ close parenthesis$

$10 x^{2} + 96 x + 180 > 17.2$

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.

$x squared greater than 12.2$

$x less than minus 3.5$ or $x greater than 3.5$

$x squared less than or equal to 45$

$- 3 x^{2} + 7 x - 25 \leq 0$

All $x$

$2.4 x^{2} - 5.6 x + 18 \leq 0$

$0.4 x squared minus 54 x less than 620$

$minus 10.6 less than x less than 145.6$

$- 0.05 x^{2} - 3 x > 76$

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

$(x + 3) (x - 4) < 0$

$open parenthesis minus 3, 4 close parenthesis$

$(x + 2) (x + 5) > 0$

$28 - 3 x - x^{2} \geq 0$

$[ minus 7, 4 ]$

$32 + 4 x - x^{2} \leq 0$

$2 z squared minus 7 z greater than 4$

$(- \infty, \frac{- 1}{2}) \cup (4, \infty)$

$6 h squared plus 13 h less than 15$

$64 minus t squared greater than 0$

$open parenthesis minus 8, 8 close parenthesis$

$121 minus y squared less than 0$

$v squared less than 5$

$open parenthesis minus 2.24, 2.24 close parenthesis$

$t squared greater than or equal to 7$

$5 a^{2} - 32 a + 12 \geq 0$

$open parenthesis minus ∞, 0.4 ] union [ 6, ∞ close parenthesis$

$6 b^{2} + 16 b - 9 < 0$

$4 x^{2} + x \geq - 2 x^{2} + 2$

$(- \infty, - 0.67) \cup (0.5, \infty)$

$2 x^{2} + 8 x \leq - x^{2} + 3$

$x^{2} - 4 x + 1 \geq 0$

$open parenthesis minus ∞, 0.27 ] union [ 3.73, ∞ close parenthesis$

$x^{2} + 4 x + 2 \leq 0$

$- 3 - m^{2} < 0$

All $m$

$11 plus n squared less than 0$

$w^{2} - w + 4 \leq 0$

No solution

$- z^{2} + z - 1 \leq 0$

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 $t$ seconds after launch is given by

$h = 320 t - 16 t^{2}$

During what time interval is the rocket higher than $1024$ feet?

$320 t minus 16 t squared greater than 1024$ ; $4 less than t less than 16$ : Between $4$ and $16$ seconds

A baseball thrown vertically reaches a height, $h$ , in feet given by

$h = 56 t - 16 t^{2}$

where $t$ is measured in seconds. During what time intervals is the ball between $40$ and $48$ feet high?

The cost, in dollars, of manufacturing $x$ pairs of garden shears is given by the function

$C = - 0.02 x^{2} + 14 x + 1600$

for $0 less than or equal to x less than or equal to 700$ . How many pairs of shears can be produced if the total cost must be kept under $ $2800$ ?

$- 0.02 x^{2} + 14 x + 1600 < 2800$ ; $[ 0, 100 close parenthesis union open parenthesis 600, 700 ]$ : Either less than $100$ or more than $600$ shears.

The cost, in dollars, of producing $x$ cashmere sweaters is given by the function

$C = x^{2} + 4 x + 90$

How many sweaters can be produced if the total cost must be kept under $$ 1850$ ?

The Locker Room finds that it sells $1200 minus 30 p$ sweatshirts each month when it charges $p$ dollars per sweatshirt. It would like its revenue from sweatshirts to be over $ $9000$ per month. In what interval should it keep the price of a sweatshirt?

$p open parenthesis 1200 minus 30 p close parenthesis greater than 9000$ ; $10 less than p less than 30$ : Between $ $10$ and $ $30$

Green Valley Nursery sells $120 minus 10 p$ boxes of rose food per month at a price of $p$ dollars per box. It would like to keep its monthly revenue from rose food over $ $350$ . In what interval should it price a box of rose food?

A group of cylindrical storage tanks must be $20$ feet tall. If the volume of each tank must be between $500 π$ and $2880 π$ cubic feet, what are the possible values for the radius of a tank?

$500 π less than 20 π r squared less than 2880 π$ ; $5 less than r less than 12$ ; The radius must be between $5$ and $12$ ft.

The volume of a cylindrical can should be between $21.2$ and $21.6$ cubic inches. If the height of the can is $5$ 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 $ $600$ per person for a weekend in Lake Tahoe if $20$ people sign up. For each additional person who signs up, the price for all participants is reduced by $ $10$ per person.

Write algebraic expressions for the size of the group and the price per person if $x$ additional people sign up.
Write a formula for the travel agency's total income as a function of $x$ .
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 $ $15, 750$ ?
Graph the income function and use the graph to verify your answers to parts (c) and (d).

Size of group: $20 plus x$ ; Price per person: $600 minus 10 x$
$I = (20 + x) (600 - 10 x)$
$ $16, 000$ ; $20$
Between 5 and 35

A farmer inherits an apple orchard on which $60$ 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 $\frac{1}{2}$ bushel.

Write algebraic expressions for the number of trees per acre and for the yield per tree if $x$ trees per acre are removed.
Write a formula for the total yield per acre as a function of $x$ .
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).

Modeling, Functions, and Graphs by Katherine Yoshiwara (yoshiwarabooks.org), GNU Free Documentation License 1.2 or later. Adapted for the XYZ HTML edition with the authors' permission (recorded 2026-07-04). License: GFDL-1.2-or-later.