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== 5.4#h(0.6em)Solve Equations with Decimals

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Evaluate #math.equation(block: false, alt: "x plus the fraction 2 over 3 when x equals minus the fraction 1 over 4 .")[$x + frac(2, 3) #h(0.2em) "when" #h(0.2em) x = − frac(1, 4) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 over 12")[$frac(5, 12)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate #math.equation(block: false, alt: "15 minus y")[$15 − y$] when #math.equation(block: false, alt: "y equals −5 .")[$y = −5 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "20")[$20$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve #math.equation(block: false, alt: "the fraction n over minus 7 equals 42 .")[$frac(n, − 7) = 42 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "minus 294")[$− 294$]
]
]


=== Determine Whether a Decimal is a Solution of an Equation

Solving equations with decimals is important in our everyday lives because money is usually written with decimals. When applications involve money, such as shopping for yourself, making your family’s budget, or planning for the future of your business, you’ll be solving equations with decimals.

Now that we’ve worked with decimals, we are ready to find solutions to equations involving decimals. The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, a fraction, or a decimal. We’ll list these steps here again for easy reference.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Determine whether a number is a solution to an equation.]

+ Substitute the number for the variable in the equation.
+ Simplify the expressions on both sides of the equation.
+ Determine whether the resulting equation is true.
- If so, the number is a solution.
- If not, the number is not a solution.
]

#examplebox("Example 1")[][
Determine whether each of the following is a solution of #math.equation(block: false, alt: "x minus 0.7 equals 1.5 :")[$x − 0.7 = 1.5 ":"$]

ⓐ #math.equation(block: false, alt: "x equals 1")[$#h(0.2em) x = 1$]ⓑ #math.equation(block: false, alt: "x equals −0.8")[$#h(0.2em) x = −0.8$]ⓒ #math.equation(block: false, alt: "x equals 2.2")[$#h(0.2em) x = 2.2$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓐ], []),
  [], [#figure(figph[The image shows the mathematical equation x - 0.7 = 1.5, presented in a clear, sans-serif font. The equation is horizontally centered against a white background.], alt: "The image shows the mathematical equation x - 0.7 = 1.5, presented in a clear, sans-serif font. The equation is horizontally centered against a white background.", caption: none)],
  [#figure(figph[The text 'Substitute 1 for x.' is shown in a dark teal color with the number 1 highlighted in red.], alt: "The text 'Substitute 1 for x.' is shown in a dark teal color with the number 1 highlighted in red.", caption: none)], [#figure(figph[A mathematical expression shows '1 - 0.7' on the left side, followed by an equals sign with a question mark above it, and '1.5' on the right side. The number '1' is highlighted in red.], alt: "A mathematical expression shows '1 - 0.7' on the left side, followed by an equals sign with a question mark above it, and '1.5' on the right side. The number '1' is highlighted in red.", caption: none)],
  [Subtract.], [#figure(figph[A white background displays the mathematical expression '0.3 ≠ 1.5' in black text, indicating that 0.3 is not equal to 1.5.], alt: "A white background displays the mathematical expression '0.3 ≠1.5' in black text, indicating that 0.3 is not equal to 1.5.", caption: none)],
))

Since #math.equation(block: false, alt: "x equals 1")[$x = 1$] does not result in a true equation, #math.equation(block: false, alt: "1")[$1$] is not a solution to the equation.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓑ], []),
  [], [#figure(figph[A mathematical equation is displayed, showing 'x - 0.7 = 1.5' in black text against a white background.], alt: "A mathematical equation is displayed, showing 'x - 0.7 = 1.5' in black text against a white background.", caption: none)],
  [#figure(figph[Substitute -0.8 for x.], alt: "Substitute -0.8 for x.", caption: none)], [#figure(figph[A mathematical expression displays -0.8 (in red) - 0.7, followed by an equals sign with a question mark on top, and then 1.5, posing whether the two sides are equal.], alt: "A mathematical expression displays -0.8 (in red) - 0.7, followed by an equals sign with a question mark on top, and then 1.5, posing whether the two sides are equal.", caption: none)],
  [Subtract.], [#figure(figph[A mathematical expression showing that -1.5 is not equal to 1.5, displayed in black text on a white background.], alt: "A mathematical expression showing that -1.5 is not equal to 1.5, displayed in black text on a white background.", caption: none)],
))

Since #math.equation(block: false, alt: "x equals −0.8")[$x = −0.8$] does not result in a true equation, #math.equation(block: false, alt: "−0.8")[$−0.8$] is not a solution to the equation.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([ⓒ], []),
  [], [#figure(figph[A mathematical equation is displayed on a white background, reading 'x - 0.7 = 1.5'.], alt: "A mathematical equation is displayed on a white background, reading 'x - 0.7 = 1.5'.", caption: none)],
  [#figure(figph[The text 'Substitute 2.2 for x.' is shown, with the number '2.2' highlighted in red.], alt: "The text 'Substitute 2.2 for x.' is shown, with the number '2.2' highlighted in red.", caption: none)], [#figure(figph[The expression 2.2 - 0.7 ?= 1.5, testing whether the subtraction of decimals results in 1.5. The number 2.2 is highlighted in red.], alt: "The expression 2.2 - 0.7 ?= 1.5, testing whether the subtraction of decimals results in 1.5. The number 2.2 is highlighted in red.", caption: none)],
  [Subtract.], [#figure(figph[The equation 1.5 = 1.5 is shown with a checkmark, indicating it is correct or verified.], alt: "The equation 1.5 = 1.5 is shown with a checkmark, indicating it is correct or verified.", caption: none)],
))

Since #math.equation(block: false, alt: "x equals 2.2")[$x = 2.2$] results in a true equation, #math.equation(block: false, alt: "2.2")[$2.2$] is a solution to the equation.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Determine whether each value is a solution of the given equation.

#math.equation(block: false, alt: "x minus 0.6 equals 1.3 :")[$x − 0.6 = 1.3 : #h(0.2em)$]ⓐ #math.equation(block: false, alt: "x equals 0.7")[$#h(0.2em) x = 0.7$]ⓑ #math.equation(block: false, alt: "x equals 1.9")[$#h(0.2em) x = 1.9$]ⓒ #math.equation(block: false, alt: "x equals −0.7")[$#h(0.2em) x = −0.7$]

#solutionbox[
+ ⓐ no
+ ⓑ yes
+ ⓒ no
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Determine whether each value is a solution of the given equation.

#math.equation(block: false, alt: "y minus 0.4 equals 1.7 :")[$y − 0.4 = 1.7 : #h(0.2em)$]ⓐ #math.equation(block: false, alt: "y equals 2.1")[$#h(0.2em) y = 2.1$]ⓑ #math.equation(block: false, alt: "y equals 1.3")[$#h(0.2em) y = 1.3$]ⓒ #math.equation(block: false, alt: "−1.3")[$#h(0.2em) −1.3$]

#solutionbox[
+ ⓐ yes
+ ⓑ no
+ ⓒ no
]
]


=== Solve Equations with Decimals

In previous chapters, we solved equations using the Properties of Equality. We will use these same properties to solve equations with decimals.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Properties of Equality]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#strong[Subtraction Property of Equality] #linebreak() For any numbers #math.equation(block: false, alt: "a , b , and c ,")[$a , b , #h(0.2em) "and" #h(0.2em) c ,$] #linebreak() If #math.equation(block: false, alt: "a equals b ,")[$a = b ,$] then #math.equation(block: false, alt: "a minus c equals b minus c .")[$a − c = b − c .$]], [#strong[Addition Property of Equality] #linebreak() 
For any numbers #math.equation(block: false, alt: "a , b , and c ,")[$a , b , #h(0.2em) "and" #h(0.2em) c ,$] #linebreak() If #math.equation(block: false, alt: "a equals b ,")[$a = b ,$] then #math.equation(block: false, alt: "a plus c equals b plus c .")[$a + c = b + c .$]]),
  [#strong[The Division Property of Equality] #linebreak() 
For any numbers #math.equation(block: false, alt: "a , b , and c , and c not equal to 0")[$a , b , #h(0.2em) "and" #h(0.2em) c , #h(0.2em) "and" #h(0.2em) c ≠ 0$] #linebreak() If #math.equation(block: false, alt: "a equals b ,")[$a = b ,$] then #math.equation(block: false, alt: "the fraction a over c equals the fraction b over c")[$frac(a, c) = frac(b, c)$]], [#strong[The Multiplication Property of Equality] #linebreak() 
For any numbers #math.equation(block: false, alt: "a , b , and c ,")[$a , b , #h(0.2em) "and" #h(0.2em) c ,$] #linebreak() If #math.equation(block: false, alt: "a equals b ,")[$a = b ,$] then #math.equation(block: false, alt: "a c equals b c")[$a c = b c$]],
))
]

When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.

#examplebox("Example 2")[][
Solve: #math.equation(block: false, alt: "y plus 2.3 equals −4.7 .")[$y + 2.3 = −4.7 .$]

#solutionbox[
We will use the #strong[Subtraction Property of Equality] to isolate the variable. #linebreak()

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation is displayed: y + 2.3 = -4.7. This is a linear equation with one variable 'y' that requires solving for 'y' by isolating it on one side of the equation.], alt: "A mathematical equation is displayed: y + 2.3 = -4.7. This is a linear equation with one variable 'y' that requires solving for 'y' by isolating it on one side of the equation.", caption: none)], [],
  [#figure(figph[The text reads 'Subtract 2.3 from each side, to undo the addition.'], alt: "The text reads 'Subtract 2.3 from each side, to undo the addition.'", caption: none)], [#figure(figph[A mathematical equation illustrating the process of solving for 'y' by subtracting 2.3 from both sides: y + 2.3 - 2.3 = -4.7 - 2.3, with the subtracted 2.3 highlighted in red.], alt: "A mathematical equation illustrating the process of solving for 'y' by subtracting 2.3 from both sides: y + 2.3 - 2.3 = -4.7 - 2.3, with the subtracted 2.3 highlighted in red.", caption: none)], [],
  [Simplify.], [#figure(figph[The image displays the equation y = -7 in black text on a white background, representing a horizontal line in a coordinate system.], alt: "The image displays the equation y = -7 in black text on a white background, representing a horizontal line in a coordinate system.", caption: none)], [],
  [#strong[Check:]], [#figure(figph[A mathematical equation displayed on a white background, which reads 'y + 2.3 = -4.7'.], alt: "A mathematical equation displayed on a white background, which reads 'y + 2.3 = -4.7'.", caption: none)], [],
  [#figure(figph[The image shows the text 'Substitute y = -7.' in a mathematical context, indicating an instruction to replace the variable 'y' with the value -7.], alt: "The image shows the text 'Substitute y = -7.' in a mathematical context, indicating an instruction to replace the variable 'y' with the value -7.", caption: none)], [#figure(figph[A mathematical equation checks if -7 + 2.3 equals -4.7, which is true.], alt: "A mathematical equation checks if -7 + 2.3 equals -4.7, which is true.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical expression shows '-4.7 = -4.7' followed by a checkmark, indicating the equality is correct.], alt: "A mathematical expression shows '-4.7 = -4.7' followed by a checkmark, indicating the equality is correct.", caption: none)], [],
))

Since #math.equation(block: false, alt: "y equals −7")[$y = −7$] makes #math.equation(block: false, alt: "y plus 2.3 equals −4.7")[$y + 2.3 = −4.7$] a true statement, we know we have found a solution to this equation.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "y plus 2.7 equals −5.3 .")[$y + 2.7 = −5.3 .$]

#solutionbox[
#emph[y] = −8
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "y plus 3.6 equals −4.8 .")[$y + 3.6 = −4.8 .$]

#solutionbox[
#emph[y] = −8.4
]
]

#examplebox("Example 3")[][
Solve: #math.equation(block: false, alt: "a minus 4.75 equals −1.39 .")[$a − 4.75 = −1.39 .$]

#solutionbox[
We will use the Addition Property of Equality. #linebreak()

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation is displayed, showing 'a - 4.75 = -1.39' against a white background.], alt: "A mathematical equation is displayed, showing 'a - 4.75 = -1.39' against a white background.", caption: none)], [],
  [Add 4.75 to each side, to undo the subtraction.], [#figure(figph[A mathematical equation showing the addition of 4.75 to both sides of an equation to isolate the variable 'a'. The numbers added are highlighted in red.], alt: "A mathematical equation showing the addition of 4.75 to both sides of an equation to isolate the variable 'a'. The numbers added are highlighted in red.", caption: none)], [],
  [Simplify.], [#figure(figph[The image displays a mathematical equation in black text on a white background, which reads 'a = 3.36'.], alt: "The image displays a mathematical equation in black text on a white background, which reads 'a = 3.36'.", caption: none)], [],
  [#strong[Check:]], [#figure(figph[A mathematical equation is displayed with the variable 'a' minus 4.75, which equals -1.39.], alt: "A mathematical equation is displayed with the variable 'a' minus 4.75, which equals -1.39.", caption: none)], [],
  [#figure(figph[The text 'Substitute], alt: "The text 'Substitute", caption: none)], [#figure(figph[A mathematical equation, '3.36 - 4.75 ?= -1.39', queries if the subtraction on the left side equals the negative value on the right. The first number, 3.36, is highlighted in red.], alt: "A mathematical equation, '3.36 - 4.75 ?= -1.39', queries if the subtraction on the left side equals the negative value on the right. The first number, 3.36, is highlighted in red.", caption: none)], [],
  [], [#figure(figph[A mathematical equation shows '-1.39 = -1.39' followed by a checkmark, indicating the equality is verified and correct.], alt: "A mathematical equation shows '-1.39 = -1.39' followed by a checkmark, indicating the equality is verified and correct.", caption: none)], [],
))

Since the result is a true statement, #math.equation(block: false, alt: "a equals 3.36")[$a = 3.36$] is a solution to the equation.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "a minus 3.93 equals −2.86 .")[$a − 3.93 = −2.86 .$]

#solutionbox[
#emph[a] = 1.07
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "n minus 3.47 equals −2.64 .")[$n − 3.47 = −2.64 .$]

#solutionbox[
#emph[n] = 0.83
]
]

#examplebox("Example 4")[][
Solve: #math.equation(block: false, alt: "−4.8 equals 0.8 n .")[$−4.8 = 0.8 n .$]

#solutionbox[
We will use the Division Property of Equality.

Use the Properties of Equality to find a value for #math.equation(block: false, alt: "n .")[$n .$] #linebreak()

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation shows '-4.8 = 0.8n' against a white background.], alt: "A mathematical equation shows '-4.8 = 0.8n' against a white background.", caption: none)], [],
  [We must divide both sides by 0.8 to isolate #emph[n].], [#figure(figph[The equation -4.8/0.8 = 0.8n/0.8, demonstrating division by 0.8 on both sides to solve for 'n', with the divisor highlighted in red.], alt: "The equation -4.8/0.8 = 0.8n/0.8, demonstrating division by 0.8 on both sides to solve for 'n', with the divisor highlighted in red.", caption: none)], [],
  [Simplify.], [#figure(figph[-6 = n, an algebraic equation where the variable n is equal to negative six.], alt: "-6 = n, an algebraic equation where the variable n is equal to negative six.", caption: none)], [],
  [#strong[Check:]], [#figure(figph[An image showing the algebraic equation -4.8 = 0.8n, which involves a negative decimal, an equals sign, another decimal, and the variable 'n'.], alt: "An image showing the algebraic equation -4.8 = 0.8n, which involves a negative decimal, an equals sign, another decimal, and the variable 'n'.", caption: none)], [],
  [#figure(figph[The text reads 'Substitute n = -6.' on a white background. The word 'Substitute' is in dark teal, 'n = ' is in dark teal, and '-6.' is in red.], alt: "The text reads 'Substitute n = -6.' on a white background. The word 'Substitute' is in dark teal, 'n = ' is in dark teal, and '-6.' is in red.", caption: none)], [#figure(figph[A mathematical equation asks whether -4.8 is equal to 0.8 multiplied by -6, with the -6 highlighted in red. The equality holds true as 0.8 \* -6 also equals -4.8.], alt: "A mathematical equation asks whether -4.8 is equal to 0.8 multiplied by -6, with the -6 highlighted in red. The equality holds true as 0.8 * -6 also equals -4.8.", caption: none)], [],
  [], [#figure(figph[The image displays the equation '-4.8 = -4.8' followed by a black checkmark, indicating that the equality is confirmed or correct.], alt: "The image displays the equation '-4.8 = -4.8' followed by a black checkmark, indicating that the equality is confirmed or correct.", caption: none)], [],
))

Since #math.equation(block: false, alt: "n equals −6")[$n = −6$] makes #math.equation(block: false, alt: "−4.8 equals 0.8 n")[$−4.8 = 0.8 n$] a true statement, we know we have a solution.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "−8.4 equals 0.7 b .")[$−8.4 = 0.7 b .$]

#solutionbox[
#emph[b] = −12
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "−5.6 equals 0.7 c .")[$−5.6 = 0.7 c .$]

#solutionbox[
#emph[c] = −8
]
]

#examplebox("Example 5")[][
Solve: #math.equation(block: false, alt: "the fraction p over minus 1.8 equals −6.5 .")[$frac(p, − 1.8) = −6.5 .$]

#solutionbox[
We will use the #strong[Multiplication Property of Equality]. #linebreak()

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[The image displays the algebraic equation p divided by -1.8 equals -6.5, presented in a black font against a white background.], alt: "The image displays the algebraic equation p divided by -1.8 equals -6.5, presented in a black font against a white background.", caption: none)], [],
  [Here, #emph[p] is divided by −1.8. We must multiply by −1.8 to isolate #emph[p]], [#figure(figph[The mathematical equation -1.8(p/-1.8) = -1.8(-6.5) is shown.], alt: "The mathematical equation -1.8(p/-1.8) = -1.8(-6.5) is shown.", caption: none)], [],
  [Multiply.], [#figure(figph[The variable 'p' is assigned the numerical value of 11.7, displayed in a simple mathematical expression.], alt: "The variable 'p' is assigned the numerical value of 11.7, displayed in a simple mathematical expression.", caption: none)], [],
  [#strong[Check:]], [#figure(figph[A mathematical equation shows p divided by -1.8, which equals -6.5.], alt: "A mathematical equation shows p divided by -1.8, which equals -6.5.", caption: none)], [],
  [#figure(figph[The text reads 'Substitute p = 11.7.', instructing to replace the variable p with the numerical value 11.7 in a mathematical context. The number 11.7 is highlighted in red.], alt: "The text reads 'Substitute p = 11.7.', instructing to replace the variable p with the numerical value 11.7 in a mathematical context. The number 11.7 is highlighted in red.", caption: none)], [#figure(figph[A mathematical equation questions if the division of 11.7 by -1.8 is equal to -6.5.], alt: "A mathematical equation questions if the division of 11.7 by -1.8 is equal to -6.5.", caption: none)], [],
  [], [#figure(figph[The equation -6.5 = -6.5 is displayed, followed by a checkmark, indicating the equality is correct.], alt: "The equation -6.5 = -6.5 is displayed, followed by a checkmark, indicating the equality is correct.", caption: none)], [],
))

A solution to #math.equation(block: false, alt: "the fraction p over −1.8 equals −6.5")[$frac(p, −1.8) = −6.5$] is #math.equation(block: false, alt: "p equals 11.7 .")[$p = 11.7 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction c over −2.6 equals −4.5 .")[$frac(c, −2.6) = −4.5 .$]

#solutionbox[
#emph[c] = 11.7
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction b over −1.2 equals −5.4 .")[$frac(b, −1.2) = −5.4 .$]

#solutionbox[
#emph[b] = 6.48
]
]


=== Translate to an Equation and Solve

Now that we have solved equations with decimals, we are ready to translate word sentences to equations and solve. Remember to look for words and phrases that indicate the operations to use.

#examplebox("Example 6")[][
Translate and solve: The difference of #math.equation(block: false, alt: "n")[$n$] and #math.equation(block: false, alt: "4.3")[$4.3$] is #math.equation(block: false, alt: "2.1 .")[$2.1 .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [Translate.], [#figure(figph[Demonstration of translating the word phrase 'The difference of n and 4.3 is 2.1' into the mathematical equation 'n - 4.3 = 2.1', using brackets for visual mapping.], alt: "Demonstration of translating the word phrase 'The difference of n and 4.3 is 2.1' into the mathematical equation 'n - 4.3 = 2.1', using brackets for visual mapping.", caption: none)], [],
  [Add #math.equation(block: false, alt: "4.3")[$4.3$] to both sides of the equation.], [#figure(figph[Algebraic solution step: Adding 4.3 to both sides of the equation n - 4.3 = 2.1 to solve for n. The red numbers highlight the operation.], alt: "Algebraic solution step: Adding 4.3 to both sides of the equation n - 4.3 = 2.1 to solve for n. The red numbers highlight the operation.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical expression on a white background displays 'n = 6.4' in bold, black text.], alt: "A mathematical expression on a white background displays 'n = 6.4' in bold, black text.", caption: none)], [],
  [#strong[Check:]], [Is the difference of #math.equation(block: false, alt: "n")[$n$] and 4.3 equal to 2.1?], [],
  [Let #math.equation(block: false, alt: "n equals 6.4")[$n = 6.4$]:], [Is the difference of 6.4 and 4.3 equal to 2.1?], [],
  [Translate.], [#figure(figph[A mathematical expression displaying '6.4 - 4.3' with a small '2' superscript on '4.3', followed by a symbol resembling an equality sign also with a small '2' superscript, then '2.1'.], alt: "A mathematical expression displaying '6.4 - 4.3' with a small '2' superscript on '4.3', followed by a symbol resembling an equality sign also with a small '2' superscript, then '2.1'.", caption: none)], [],
  [Simplify.], [#figure(figph[The equation 2.1 = 2.1 is shown with a checkmark, indicating that the mathematical statement is correct.], alt: "The equation 2.1 = 2.1 is shown with a checkmark, indicating that the mathematical statement is correct.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: The difference of #math.equation(block: false, alt: "y")[$y$] and #math.equation(block: false, alt: "4.9")[$4.9$] is #math.equation(block: false, alt: "2.8 .")[$2.8 .$]

#solutionbox[
#emph[y] − 4.9 = 2.8; #emph[y] = 7.7
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: The difference of #math.equation(block: false, alt: "z")[$z$] and #math.equation(block: false, alt: "5.7")[$5.7$] is #math.equation(block: false, alt: "3.4 .")[$3.4 .$]

#solutionbox[
#emph[z] − 5.7 = 3.4; #emph[z] = 9.1
]
]

#examplebox("Example 7")[][
Translate and solve: The product of #math.equation(block: false, alt: "−3.1")[$−3.1$] and #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "5.27 .")[$5.27 .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [Translate.], [#figure(figph[An image translates 'The product of 3.1 and x is 5.27' into the algebraic equation '-3.1x = 5.27', showing an unexpected negative sign in the term -3.1x compared to the verbal statement.], alt: "An image translates 'The product of 3.1 and x is 5.27' into the algebraic equation '-3.1x = 5.27', showing an unexpected negative sign in the term -3.1x compared to the verbal statement.", caption: none)], [],
  [Divide both sides by #math.equation(block: false, alt: "−3.1")[$−3.1$].], [#figure(figph[A step in solving an algebraic equation, showing both sides of the equation -3.1x = 5.27 being divided by -3.1, highlighting the division operation to isolate x.], alt: "A step in solving an algebraic equation, showing both sides of the equation -3.1x = 5.27 being divided by -3.1, highlighting the division operation to isolate x.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical expression 'x = -1.7' is displayed on a plain white background.], alt: "A mathematical expression 'x = -1.7' is displayed on a plain white background.", caption: none)], [],
  [#strong[Check:]], [Is the product of −3.1 and #math.equation(block: false, alt: "x")[$x$] equal to #math.equation(block: false, alt: "5.27")[$5.27$]?], [],
  [Let #math.equation(block: false, alt: "x equals −1.7")[$x = −1.7$]:], [Is the product of #math.equation(block: false, alt: "−3.1")[$−3.1$] and #math.equation(block: false, alt: "−1.7")[$−1.7$] equal to #math.equation(block: false, alt: "5.27")[$5.27$]?], [],
  [Translate.], [#figure(figph[A mathematical problem displaying the equation -3.1(-1.7)?=5.27, which asks to determine if -3.1 multiplied by -1.7 indeed equals 5.27.], alt: "A mathematical problem displaying the equation -3.1(-1.7)?=5.27, which asks to determine if -3.1 multiplied by -1.7 indeed equals 5.27.", caption: none)], [],
  [Simplify.], [#figure(figph[The equation 5.27 = 5.27 with a checkmark, indicating correctness.], alt: "The equation 5.27 = 5.27 with a checkmark, indicating correctness.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: The product of #math.equation(block: false, alt: "−4.3")[$−4.3$] and #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "12.04 .")[$12.04 .$]

#solutionbox[
−4.3#emph[x] = 12.04; #emph[x] = −2.8
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: The product of #math.equation(block: false, alt: "−3.1")[$−3.1$] and #math.equation(block: false, alt: "m")[$m$] is #math.equation(block: false, alt: "26.66 .")[$26.66 .$]

#solutionbox[
−3.1#emph[m] = 26.66; #emph[m] = −8.6
]
]

#examplebox("Example 8")[][
Translate and solve: The quotient of #math.equation(block: false, alt: "p")[$p$] and #math.equation(block: false, alt: "−2.4")[$−2.4$] is #math.equation(block: false, alt: "6.5 .")[$6.5 .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [Translate.], [#figure(figph[This image translates the verbal statement 'The quotient of p and -2.4 is 6.5' into the mathematical equation p / -2.4 = 6.5, visually connecting each part of the phrase to its symbolic representation.], alt: "This image translates the verbal statement 'The quotient of p and -2.4 is 6.5' into the mathematical equation p / -2.4 = 6.5, visually connecting each part of the phrase to its symbolic representation.", caption: none)], [],
  [Multiply both sides by #math.equation(block: false, alt: "−2.4")[$−2.4$].], [#figure(figph[A mathematical equation shows '-2.4(P/-2.4) = -2.4(6.5)'. The number -2.4 is highlighted in red, indicating multiplication by -2.4 on both sides of the equation.], alt: "A mathematical equation shows '-2.4(P/-2.4) = -2.4(6.5)'. The number -2.4 is highlighted in red, indicating multiplication by -2.4 on both sides of the equation.", caption: none)], [],
  [Simplify.], [#figure(figph[The image displays a mathematical equation: p = -15.6, rendered in a black serif font on a plain white background.], alt: "The image displays a mathematical equation: p = -15.6, rendered in a black serif font on a plain white background.", caption: none)], [],
  [#strong[Check:]], [Is the quotient of #math.equation(block: false, alt: "p")[$p$] and #math.equation(block: false, alt: "−2.4")[$−2.4$] equal to #math.equation(block: false, alt: "6.5")[$6.5$]?], [],
  [Let #math.equation(block: false, alt: "p equals −15.6 :")[$p = −15.6 :$]], [Is the quotient of #math.equation(block: false, alt: "−15.6")[$−15.6$] and #math.equation(block: false, alt: "−2.4")[$−2.4$] equal to #math.equation(block: false, alt: "6.5")[$6.5$]?], [],
  [Translate.], [#figure(figph[A mathematical equation questions whether -15.6 divided by -2.4 is approximately equal to 6.5.], alt: "A mathematical equation questions whether -15.6 divided by -2.4 is approximately equal to 6.5.", caption: none)], [],
  [Simplify.], [#figure(figph[The equation 6.5 = 6.5 is shown with a checkmark, indicating that the equality is correct or verified.], alt: "The equation 6.5 = 6.5 is shown with a checkmark, indicating that the equality is correct or verified.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: The quotient of #math.equation(block: false, alt: "q")[$q$] and #math.equation(block: false, alt: "−3.4")[$−3.4$] is #math.equation(block: false, alt: "4.5 .")[$4.5 .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction q over −3.4 equals 4.5 ; q equals −15.3")[$frac(q, −3.4) = 4.5 ; #h(0.2em) q = −15.3$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: The quotient of #math.equation(block: false, alt: "r")[$r$] and #math.equation(block: false, alt: "−2.6")[$−2.6$] is #math.equation(block: false, alt: "2.5 .")[$2.5 .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction r over −2.6 equals 2.5 ; r equals −6.5")[$frac(r, −2.6) = 2.5 ; #h(0.2em) r = −6.5$]
]
]

#examplebox("Example 9")[][
Translate and solve: The sum of #math.equation(block: false, alt: "n")[$n$] and #math.equation(block: false, alt: "2.9")[$2.9$] is #math.equation(block: false, alt: "1.7 .")[$1.7 .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [Translate.], [#figure(figph[An image explaining how to translate a word problem into an algebraic equation. It shows 'The sum of n and 2.9 is 1.7.' translated to 'n + 2.9 = 1.7', illustrating the breakdown of the sentence into its mathematical components.], alt: "An image explaining how to translate a word problem into an algebraic equation. It shows 'The sum of n and 2.9 is 1.7.' translated to 'n + 2.9 = 1.7', illustrating the breakdown of the sentence into its mathematical components.", caption: none)], [],
  [Subtract #math.equation(block: false, alt: "2.9")[$2.9$] from each side.], [#figure(figph[An algebraic equation showing the step of subtracting 2.9 from both sides: n + 2.9 - 2.9 = 1.7 - 2.9, with the subtracted 2.9 highlighted in red.], alt: "An algebraic equation showing the step of subtracting 2.9 from both sides: n + 2.9 - 2.9 = 1.7 - 2.9, with the subtracted 2.9 highlighted in red.", caption: none)], [],
  [Simplify.], [#figure(figph[The image displays the mathematical equation 'n = -1.2' in black text on a plain white background, occupying the top right portion of the frame.], alt: "The image displays the mathematical equation 'n = -1.2' in black text on a plain white background, occupying the top right portion of the frame.", caption: none)], [],
  [#strong[Check:]], [Is the sum #math.equation(block: false, alt: "n")[$n$] and #math.equation(block: false, alt: "2.9")[$2.9$] equal to #math.equation(block: false, alt: "1.7")[$1.7$]?], [],
  [Let #math.equation(block: false, alt: "n equals −1.2 :")[$n = −1.2 :$]], [Is the sum #math.equation(block: false, alt: "−1.2")[$−1.2$] and #math.equation(block: false, alt: "2.9")[$2.9$] equal to #math.equation(block: false, alt: "1.7")[$1.7$]?], [],
  [Translate.], [#figure(figph[A mathematical equation reads '-1.2 + 2.9 =? 1.7', showing the addition of two decimal numbers with a question mark next to the equals sign, suggesting a verification or problem to solve.], alt: "A mathematical equation reads '-1.2 + 2.9 =? 1.7', showing the addition of two decimal numbers with a question mark next to the equals sign, suggesting a verification or problem to solve.", caption: none)], [],
  [Simplify.], [#figure(figph[The image shows the mathematical equality '1.7 = 1.7' followed by a checkmark, indicating the correctness of the statement.], alt: "The image shows the mathematical equality '1.7 = 1.7' followed by a checkmark, indicating the correctness of the statement.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: The sum of #math.equation(block: false, alt: "j")[$j$] and #math.equation(block: false, alt: "3.8")[$3.8$] is #math.equation(block: false, alt: "2.6 .")[$2.6 .$]

#solutionbox[
#emph[j] + 3.8 = 2.6; #emph[j] = −1.2
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: The sum of #math.equation(block: false, alt: "k")[$k$] and #math.equation(block: false, alt: "4.7")[$4.7$] is #math.equation(block: false, alt: "0.3 .")[$0.3 .$]

#solutionbox[
#emph[k] + 4.7 = 0.3; #emph[k] = −4.4
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[ACCESS ADDITIONAL ONLINE RESOURCES]

- #link("https://openstaxcollege.org/l/24eqwithdec")[Solving One Step Equations Involving Decimals]
- #link("https://openstaxcollege.org/l/24eqnwdecplsmin")[Solve a One Step Equation With Decimals by Adding and Subtracting]
- #link("https://openstaxcollege.org/l/24eqnwdecmult")[Solve a One Step Equation With Decimals by Multiplying]
- #link("https://openstaxcollege.org/l/24eqnwdecdiv")[Solve a One Step Equation With Decimals by Dividing]
]


=== Key Concepts

- #emph[Determine whether a number is a solution to an equation.]
- Substitute the number for the variable in the equation.
- Simplify the expressions on both sides of the equation.
- Determine whether the resulting equation is true. #linebreak() 
If so, the number is a solution. #linebreak() 
If not, the number is not a solution.
- #emph[Properties of Equality]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#emph[Subtraction Property of Equality]], [#emph[Addition Property of Equality]]),
  [For any numbers #math.equation(block: false, alt: "a")[$a$], #math.equation(block: false, alt: "b")[$b$], and #math.equation(block: false, alt: "c")[$c$], #linebreak() 
#math.equation(block: false, alt: "If, a, equals, b; then, a minus c, equals, b minus c")[$"If" & a & = & b \ "then" & a − c & = & b − c$]], [For any numbers #math.equation(block: false, alt: "a")[$a$], #math.equation(block: false, alt: "b")[$b$], and #math.equation(block: false, alt: "c")[$c$], #linebreak() 
#math.equation(block: false, alt: "If, a, equals, b; then, a plus c, equals, b plus c")[$"If" & a & = & b \ "then" & a + c & = & b + c$]],
  [#emph[Division of Property of Equality]], [#emph[Multiplication Property of Equality]],
  [For any numbers #math.equation(block: false, alt: "a")[$a$], #math.equation(block: false, alt: "b")[$b$], and #math.equation(block: false, alt: "c not equal to 0")[$c ≠ 0$], #linebreak() 
#math.equation(block: false, alt: "If, a, equals, b; then, the fraction a over c, equals, the fraction b over c")[$"If" & a & = & b \ "then" & frac(a, c) & = & frac(b, c)$]], [For any numbers #math.equation(block: false, alt: "a")[$a$], #math.equation(block: false, alt: "b")[$b$], and #math.equation(block: false, alt: "c")[$c$], #linebreak() 
#math.equation(block: false, alt: "If, a, equals, b; then, a times c, equals, b times c")[$"If" & a & = & b \ "then" & a ⋅ c & = & b ⋅ c$]],
))


==== Practice Makes Perfect

#strong[Determine Whether a Decimal is a Solution of an Equation]

In the following exercises, determine whether each number is a solution of the given equation.

#math.equation(block: false, alt: "x minus 0.8 equals 2.3")[$x − 0.8 = 2.3$] #linebreak() ⓐ #math.equation(block: false, alt: "x equals 2")[$#h(0.2em) x = 2$]ⓑ #math.equation(block: false, alt: "x equals −1.5")[$#h(0.2em) x = −1.5$]ⓒ #math.equation(block: false, alt: "x equals 3.1")[$#h(0.2em) x = 3.1$]

#solutionbox[
+ ⓐ no
+ ⓑ no
+ ⓒ yes
]

#math.equation(block: false, alt: "y plus 0.6 equals −3.4")[$y + 0.6 = −3.4$] #linebreak() 
ⓐ #math.equation(block: false, alt: "y equals −4")[$#h(0.2em) y = −4$]ⓑ #math.equation(block: false, alt: "y equals −2.8")[$#h(0.2em) y = −2.8$]ⓒ #math.equation(block: false, alt: "y equals 2.6")[$#h(0.2em) y = 2.6$]

#math.equation(block: false, alt: "the fraction h over 1.5 equals −4.3")[$frac(h, 1.5) = −4.3$] #linebreak() 
ⓐ #math.equation(block: false, alt: "h equals 6.45")[$#h(0.2em) h = 6.45$]ⓑ #math.equation(block: false, alt: "h equals −6.45")[$#h(0.2em) h = −6.45$]ⓒ #math.equation(block: false, alt: "h equals −2.1")[$#h(0.2em) h = −2.1$]

#solutionbox[
+ ⓐ no
+ ⓑ yes
+ ⓒ no
]

#math.equation(block: false, alt: "0.75 k equals −3.6")[$0.75 k = −3.6$] #linebreak() 
ⓐ #math.equation(block: false, alt: "k equals −0.48")[$#h(0.2em) k = −0.48$]ⓑ #math.equation(block: false, alt: "k equals −4.8")[$#h(0.2em) k = −4.8$]ⓒ #math.equation(block: false, alt: "k equals −2.7")[$#h(0.2em) k = −2.7$]

#strong[Solve Equations with Decimals]

In the following exercises, solve the equation.

#math.equation(block: true, alt: "y plus 2.9 equals 5.7")[$y + 2.9 = 5.7$]

#solutionbox[
#emph[y] = 2.8
]

#math.equation(block: true, alt: "m plus 4.6 equals 6.5")[$m + 4.6 = 6.5$]

#math.equation(block: true, alt: "f plus 3.45 equals 2.6")[$f + 3.45 = 2.6$]

#solutionbox[
#emph[f] = −0.85
]

#math.equation(block: true, alt: "h plus 4.37 equals 3.5")[$h + 4.37 = 3.5$]

#math.equation(block: true, alt: "a plus 6.2 equals −1.7")[$a + 6.2 = −1.7$]

#solutionbox[
#emph[a] = −7.9
]

#math.equation(block: true, alt: "b plus 5.8 equals −2.3")[$b + 5.8 = −2.3$]

#math.equation(block: true, alt: "c plus 1.15 equals −3.5")[$c + 1.15 = −3.5$]

#solutionbox[
#emph[c] = −4.65
]

#math.equation(block: true, alt: "d plus 2.35 equals −4.8")[$d + 2.35 = −4.8$]

#math.equation(block: true, alt: "n minus 2.6 equals 1.8")[$n − 2.6 = 1.8$]

#solutionbox[
#emph[n] = 4.4
]

#math.equation(block: true, alt: "p minus 3.6 equals 1.7")[$p − 3.6 = 1.7$]

#math.equation(block: true, alt: "x minus 0.4 equals −3.9")[$x − 0.4 = −3.9$]

#solutionbox[
#emph[x] = −3.5
]

#math.equation(block: true, alt: "y minus 0.6 equals −4.5")[$y − 0.6 = −4.5$]

#math.equation(block: true, alt: "j minus 1.82 equals −6.5")[$j − 1.82 = −6.5$]

#solutionbox[
#emph[j] = −4.68
]

#math.equation(block: true, alt: "k minus 3.19 equals −4.6")[$k − 3.19 = −4.6$]

#math.equation(block: true, alt: "m minus 0.25 equals −1.67")[$m − 0.25 = −1.67$]

#solutionbox[
#emph[m] = −1.42
]

#math.equation(block: true, alt: "q minus 0.47 equals −1.53")[$q − 0.47 = −1.53$]

#math.equation(block: true, alt: "0.5 x equals 3.5")[$0.5 x = 3.5$]

#solutionbox[
#emph[x] = 7
]

#math.equation(block: true, alt: "0.4 p equals 9.2")[$0.4 p = 9.2$]

#math.equation(block: true, alt: "−1.7 c equals 8.5")[$−1.7 c = 8.5$]

#solutionbox[
#emph[c] = −5
]

#math.equation(block: true, alt: "−2.9 x equals 5.8")[$−2.9 x = 5.8$]

#math.equation(block: true, alt: "−1.4 p equals −4.2")[$−1.4 p = −4.2$]

#solutionbox[
#emph[p] = 3
]

#math.equation(block: true, alt: "−2.8 m equals −8.4")[$−2.8 m = −8.4$]

#math.equation(block: true, alt: "−120 equals 1.5 q")[$−120 = 1.5 q$]

#solutionbox[
#emph[q] = −80
]

#math.equation(block: true, alt: "−75 equals 1.5 y")[$−75 = 1.5 y$]

#math.equation(block: true, alt: "0.24 x equals 4.8")[$0.24 x = 4.8$]

#solutionbox[
#emph[x] = 20
]

#math.equation(block: true, alt: "0.18 n equals 5.4")[$0.18 n = 5.4$]

#math.equation(block: true, alt: "−3.4 z equals −9.18")[$−3.4 z = −9.18$]

#solutionbox[
#emph[z] = 2.7
]

#math.equation(block: true, alt: "−2.7 u equals −9.72")[$−2.7 u = −9.72$]

#math.equation(block: true, alt: "the fraction a over 0.4 equals −20")[$frac(a, 0.4) = −20$]

#solutionbox[
#emph[a] = −8
]

#math.equation(block: true, alt: "the fraction b over 0.3 equals −9")[$frac(b, 0.3) = −9$]

#math.equation(block: true, alt: "the fraction x over 0.7 equals −0.4")[$frac(x, 0.7) = −0.4$]

#solutionbox[
#emph[x] = −0.28
]

#math.equation(block: true, alt: "the fraction y over 0.8 equals −0.7")[$frac(y, 0.8) = −0.7$]

#math.equation(block: true, alt: "the fraction p over minus 5 equals −1.65")[$frac(p, − 5) = −1.65$]

#solutionbox[
#emph[p] = 8.25
]

#math.equation(block: true, alt: "the fraction q over minus 4 equals −5.92")[$frac(q, − 4) = −5.92$]

#math.equation(block: true, alt: "the fraction r over minus 1.2 equals −6")[$frac(r, − 1.2) = −6$]

#solutionbox[
#emph[r] = 7.2
]

#math.equation(block: true, alt: "the fraction s over minus 1.5 equals −3")[$frac(s, − 1.5) = −3$]

#strong[Mixed Practice]

In the following exercises, solve the equation. Then check your solution.

#math.equation(block: true, alt: "x minus 5 equals −11")[$x − 5 = −11$]

#solutionbox[
#emph[x] = −6
]

#math.equation(block: true, alt: "minus the fraction 2 over 5 equals x plus the fraction 3 over 4")[$− frac(2, 5) = x + frac(3, 4)$]

#math.equation(block: true, alt: "p plus 8 equals −2")[$p + 8 = −2$]

#solutionbox[
#emph[p] = −10
]

#math.equation(block: true, alt: "p plus the fraction 2 over 3 equals the fraction 1 over 12")[$p + frac(2, 3) = frac(1, 12)$]

#math.equation(block: true, alt: "−4.2 m equals −33.6")[$−4.2 m = −33.6$]

#solutionbox[
#emph[m] = 8
]

#math.equation(block: true, alt: "q plus 9.5 equals −14")[$q + 9.5 = −14$]

#math.equation(block: true, alt: "q plus the fraction 5 over 6 equals the fraction 1 over 12")[$q + frac(5, 6) = frac(1, 12)$]

#solutionbox[
#math.equation(block: true, alt: "q equals minus the fraction 3 over 4")[$q = − frac(3, 4)$]
]

#math.equation(block: true, alt: "the fraction 8.6 over 15 equals minus d")[$frac(8.6, 15) = − d$]

#math.equation(block: true, alt: "the fraction 7 over 8 m equals the fraction 1 over 10")[$frac(7, 8) m = frac(1, 10)$]

#solutionbox[
#math.equation(block: true, alt: "m equals the fraction 4 over 35")[$m = frac(4, 35)$]
]

#math.equation(block: true, alt: "the fraction j over minus 6.2 equals −3")[$frac(j, − 6.2) = −3$]

#math.equation(block: true, alt: "minus the fraction 2 over 3 equals y plus the fraction 3 over 8")[$− frac(2, 3) = y + frac(3, 8)$]

#solutionbox[
#math.equation(block: true, alt: "y equals minus the fraction 25 over 24")[$y = − frac(25, 24)$]
]

#math.equation(block: true, alt: "s minus 1.75 equals −3.2")[$s − 1.75 = −3.2$]

#math.equation(block: true, alt: "the fraction 11 over 20 equals minus f")[$frac(11, 20) = − f$]

#solutionbox[
#math.equation(block: true, alt: "f equals minus the fraction 11 over 20")[$f = − frac(11, 20)$]
]

#math.equation(block: true, alt: "−3.6 b equals 2.52")[$−3.6 b = 2.52$]

#math.equation(block: true, alt: "−4.2 a equals 3.36")[$−4.2 a = 3.36$]

#solutionbox[
#emph[a] = −0.8
]

#math.equation(block: true, alt: "−9.1 n equals −63.7")[$−9.1 n = −63.7$]

#math.equation(block: true, alt: "r minus 1.25 equals −2.7")[$r − 1.25 = −2.7$]

#solutionbox[
#emph[r] = −1.45
]

#math.equation(block: true, alt: "the fraction 1 over 4 n equals the fraction 7 over 10")[$frac(1, 4) n = frac(7, 10)$]

#math.equation(block: true, alt: "the fraction h over minus 3 equals −8")[$frac(h, − 3) = −8$]

#solutionbox[
#emph[h] = 24
]

#math.equation(block: true, alt: "y minus 7.82 equals −16")[$y − 7.82 = −16$]

#strong[Translate to an Equation and Solve]

In the following exercises, translate and solve.

The difference of #math.equation(block: false, alt: "n")[$n$] and #math.equation(block: false, alt: "1.9")[$1.9$] is #math.equation(block: false, alt: "3.4 .")[$3.4 .$]

#solutionbox[
#math.equation(block: true, alt: "n minus 1.9 equals 3.4 ; 5.3")[$n − 1.9 = 3.4 ; 5.3$]
]

The difference #math.equation(block: false, alt: "n")[$n$] and #math.equation(block: false, alt: "1.5")[$1.5$] is #math.equation(block: false, alt: "0.8 .")[$0.8 .$]

The product of #math.equation(block: false, alt: "−6.2")[$−6.2$] and #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "−4.96 .")[$−4.96 .$]

#solutionbox[
−6.2#emph[x] = −4.96; 0.8
]

The product of #math.equation(block: false, alt: "−4.6")[$−4.6$] and #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "−3.22 .")[$−3.22 .$]

The quotient of #math.equation(block: false, alt: "y")[$y$] and #math.equation(block: false, alt: "−1.7")[$−1.7$] is #math.equation(block: false, alt: "−5 .")[$−5 .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction y over minus 1.7 equals −5 ; 8.5")[$frac(y, − 1.7) = −5 ; #h(0.2em) 8.5$]
]

The quotient of #math.equation(block: false, alt: "z")[$z$] and #math.equation(block: false, alt: "−3.6")[$−3.6$] is #math.equation(block: false, alt: "3 .")[$3 .$]

The sum of #math.equation(block: false, alt: "n")[$n$] and #math.equation(block: false, alt: "−7.3")[$−7.3$] is #math.equation(block: false, alt: "2.4.")[$2.4.$]

#solutionbox[
#emph[n] + (−7.3) = 2.4; 9.7
]

The sum of #math.equation(block: false, alt: "n")[$n$] and #math.equation(block: false, alt: "−5.1")[$−5.1$] is #math.equation(block: false, alt: "3.8 .")[$3.8 .$]


==== Everyday Math

Shawn bought a pair of shoes on sale for #math.equation(block: false, alt: "$78")[$\$78$]. Solve the equation #math.equation(block: false, alt: "0.75 p equals 78")[$0.75 p = 78$] to find the original price of the shoes, #math.equation(block: false, alt: "p .")[$p .$]

#solutionbox[
\$104
]

Mary bought a new refrigerator. The total price including sales tax was #math.equation(block: false, alt: "$1,350 .")[$"$1,350" .$] Find the retail price, #math.equation(block: false, alt: "r ,")[$r ,$] of the refrigerator before tax by solving the equation #math.equation(block: false, alt: "1.08 r equals 1,350 .")[$1.08 r = 1,350 .$]


==== Writing Exercises

Think about solving the equation #math.equation(block: false, alt: "1.2 y equals 60 ,")[$1.2 y = 60 ,$] but do not actually solve it. Do you think the solution should be greater than #math.equation(block: false, alt: "60")[$60$] or less than #math.equation(block: false, alt: "60 ?")[$60 ?$] Explain your reasoning. Then solve the equation to see if your thinking was correct.

#solutionbox[
Answers will vary.
]

Think about solving the equation #math.equation(block: false, alt: "0.8 x equals 200 ,")[$0.8 x = 200 ,$] but do not actually solve it. Do you think the solution should be greater than #math.equation(block: false, alt: "200")[$200$] or less than #math.equation(block: false, alt: "200 ?")[$200 ?$] Explain your reasoning. Then solve the equation to see if your thinking was correct.


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

#figure(figph[A self-assessment chart for math skills related to decimals and equations. It lists three skills: determining if a decimal is a solution, solving equations with decimals, and translating to an equation and solving.], alt: "A self-assessment chart for math skills related to decimals and equations. It lists three skills: determining if a decimal is a solution, solving equations with decimals, and translating to an equation and solving.", caption: none)

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?