#set document(title: "8.3 Add and Subtract Rational Expressions with a Common Denominator", author: "OpenStax / XYZ Homework")
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== 8.3#h(0.6em)Add and Subtract Rational Expressions with a Common Denominator

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

If you miss a problem, go back to the section listed and review the material.

Add: #math.equation(block: false, alt: "the fraction y over 3 plus the fraction 9 over 3 .")[$frac(y, 3) + frac(9, 3) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction y plus 9 over 3")[$frac(y + 9, 3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Subtract: #math.equation(block: false, alt: "the fraction 10 over x minus the fraction 2 over x .")[$frac(10, x) − frac(2, x) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 8 over x")[$frac(8, x)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "8 n to the power 5 minus 20 n cubed .")[$8 n^(5) − 20 n^(3) .$] #linebreak() 
If you missed this problem, review .
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor completely: #math.equation(block: false, alt: "45 a cubed minus 5 a b squared .")[$45 a^(3) − 5 a b^(2) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "5 a 3 a minus b 3 a plus b")[$5 a 3 a − b 3 a + b$]
]
]

#linebreak()


=== Add Rational Expressions with a Common Denominator

What is the first step you take when you add numerical fractions? You check if they have a common denominator. If they do, you add the numerators and place the sum over the common denominator. If they do not have a common denominator, you find one before you add.

It is the same with rational expressions. To add rational expressions, they must have a common denominator. When the denominators are the same, you add the numerators and place the sum over the common denominator.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Rational Expression Addition]

If #math.equation(block: false, alt: "p , q , and r")[$p , q , #h(0.2em) "and" #h(0.2em) r$] are polynomials where #math.equation(block: false, alt: "r not equal to 0")[$r ≠ 0$], then

#math.equation(block: true, alt: "the fraction p over r plus the fraction q over r equals the fraction p plus q over r")[$frac(p, r) + frac(q, r) = frac(p + q, r)$]To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator.
]

We will add two numerical fractions first, to remind us of how this is done.

#examplebox("Example 1")[][
Add: #math.equation(block: false, alt: "the fraction 5 over 18 plus the fraction 7 over 18 .")[$frac(5, 18) + frac(7, 18) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 5 over 18 plus the fraction 7 over 18")[$frac(5, 18) + frac(7, 18)$]]),
  [The fractions have a common #linebreak() denominator, so add the numerators and #linebreak() place the sum over the common denominator.], [#math.equation(block: false, alt: "the fraction 5 plus 7 over 18")[$frac(5 + 7, 18)$]],
  [Add in the numerator.], [#math.equation(block: false, alt: "the fraction 12 over 18")[$frac(12, 18)$]],
  [Factor the numerator and denominator to #linebreak() show the common factors.], [#math.equation(block: false, alt: "the fraction 6 times 2 over 6 times 3")[$frac(6 · 2, 6 · 3)$]],
  [Remove common factors.], [#math.equation(block: false, alt: "the fraction 6 times 2 over 6 times 3")[$frac(cancel(6) · 2, cancel(6) · 3)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Add: #math.equation(block: false, alt: "the fraction 7 over 16 plus the fraction 5 over 16 .")[$frac(7, 16) + frac(5, 16) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Add: #math.equation(block: false, alt: "the fraction 3 over 10 plus the fraction 1 over 10 .")[$frac(3, 10) + frac(1, 10) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 over 5")[$frac(2, 5)$]
]
]

Remember, we do not allow values that would make the denominator zero. What value of #math.equation(block: false, alt: "y")[$y$] should be excluded in the next example?

#examplebox("Example 2")[][
Add: #math.equation(block: false, alt: "the fraction 3 y over 4 y minus 3 plus the fraction 7 over 4 y minus 3 .")[$frac(3 y, 4 y − 3) + frac(7, 4 y − 3) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 3 y over 4 y minus 3 plus the fraction 7 over 4 y minus 3")[$frac(3 y, 4 y − 3) + frac(7, 4 y − 3)$]]),
  [The fractions have a common #linebreak() denominator, so add the numerators and #linebreak() place the sum over the common denominator.], [#math.equation(block: false, alt: "the fraction 3 y plus 7 over 4 y minus 3")[$frac(3 y + 7, 4 y − 3)$]],
))

The numerator and denominator cannot be factored. The fraction is simplified.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Add: #math.equation(block: false, alt: "the fraction 5 x over 2 x plus 3 plus the fraction 2 over 2 x plus 3 .")[$frac(5 x, 2 x + 3) + frac(2, 2 x + 3) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Add: #math.equation(block: false, alt: "the fraction x over x minus 2 plus the fraction 1 over x minus 2 .")[$frac(x, x − 2) + frac(1, x − 2) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction x plus 1 over x minus 2")[$frac(x + 1, x − 2)$]
]
]

#examplebox("Example 3")[][
Add: #math.equation(block: false, alt: "the fraction 7 x plus 12 over x plus 3 plus the fraction x squared over x plus 3 .")[$frac(7 x + 12, x + 3) + frac(x^(2), x + 3) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 7 x plus 12 over x plus 3 plus the fraction x squared over x plus 3")[$frac(7 x + 12, x + 3) + frac(x^(2), x + 3)$]]),
  [The fractions have a common #linebreak() denominator, so add the numerators and #linebreak() place the sum over the common denominator.], [#math.equation(block: false, alt: "the fraction 7 x plus 12 plus x squared over x plus 3")[$frac(7 x + 12 + x^(2), x + 3)$]],
  [Write the degrees in descending order.], [#math.equation(block: false, alt: "the fraction x squared plus 7 x plus 12 over x plus 3")[$frac(x^(2) + 7 x + 12, x + 3)$]],
  [Factor the numerator.], [#math.equation(block: false, alt: "the fraction open parenthesis x plus 3 close parenthesis open parenthesis x plus 4 close parenthesis over x plus 3")[$frac(( x + 3 ) ( x + 4 ), x + 3)$]],
  [Simplify by removing common factors.], [#math.equation(block: false, alt: "the fraction open parenthesis x plus 3 close parenthesis open parenthesis x plus 4 close parenthesis over x plus 3")[$frac(cancel(( x + 3 )) ( x + 4 ), cancel(x + 3))$]],
  [Simplify.], [#math.equation(block: false, alt: "x plus 4")[$x + 4$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Add: #math.equation(block: false, alt: "the fraction 9 x plus 14 over x plus 7 plus the fraction x squared over x plus 7 .")[$frac(9 x + 14, x + 7) + frac(x^(2), x + 7) .$]

#solutionbox[
#math.equation(block: true, alt: "x plus 2")[$x + 2$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Add: #math.equation(block: false, alt: "the fraction x squared plus 8 x over x plus 5 plus the fraction 15 over x plus 5 .")[$frac(x^(2) + 8 x, x + 5) + frac(15, x + 5) .$]

#solutionbox[
#math.equation(block: true, alt: "x plus 3")[$x + 3$]
]
]


=== Subtract Rational Expressions with a Common Denominator

To subtract rational expressions, they must also have a common denominator. When the denominators are the same, you subtract the numerators and place the difference over the common denominator.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Rational Expression Subtraction]

If #math.equation(block: false, alt: "p , q , and r")[$p , q , #h(0.2em) "and" #h(0.2em) r$] are polynomials where #math.equation(block: false, alt: "r not equal to 0")[$r ≠ 0$], then

#math.equation(block: true, alt: "the fraction p over r minus the fraction q over r equals the fraction p minus q over r")[$frac(p, r) − frac(q, r) = frac(p − q, r)$]To subtract rational expressions, subtract the numerators and place the difference over the common denominator.
]

We always simplify rational expressions. Be sure to factor, if possible, after you subtract the numerators so you can identify any common factors.

#examplebox("Example 4")[][
Subtract: #math.equation(block: false, alt: "the fraction n squared over n minus 10 minus the fraction 100 over n minus 10 .")[$frac(n^(2), n − 10) − frac(100, n − 10) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction n squared over n minus 10 minus the fraction 100 over n minus 10")[$frac(n^(2), n − 10) − frac(100, n − 10)$]]),
  [The fractions have a common #linebreak() denominator, so subtract the numerators #linebreak() and place the difference over the common denominator.], [#math.equation(block: false, alt: "the fraction n squared minus 100 over n minus 10")[$frac(n^(2) − 100, n − 10)$]],
  [Factor the numerator.], [#math.equation(block: false, alt: "the fraction open parenthesis n minus 10 close parenthesis open parenthesis n plus 10 close parenthesis over n minus 10")[$frac(( n − 10 ) ( n + 10 ), n − 10)$]],
  [Simplify by removing common factors.], [#math.equation(block: false, alt: "the fraction open parenthesis n minus 10 close parenthesis open parenthesis n plus 10 close parenthesis over n minus 10")[$frac(cancel(( n − 10 )) ( n + 10 ), cancel(n − 10))$]],
  [Simplify.], [#math.equation(block: false, alt: "n plus 10")[$n + 10$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Subtract: #math.equation(block: false, alt: "the fraction x squared over x plus 3 minus the fraction 9 over x plus 3 .")[$frac(x^(2), x + 3) − frac(9, x + 3) .$]

#solutionbox[
#math.equation(block: true, alt: "x minus 3")[$x − 3$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Subtract: #math.equation(block: false, alt: "the fraction 4 x squared over 2 x minus 5 minus the fraction 25 over 2 x minus 5 .")[$frac(4 x^(2), 2 x − 5) − frac(25, 2 x − 5) .$]

#solutionbox[
#math.equation(block: true, alt: "2 x plus 5")[$2 x + 5$]
]
]

Be careful of the signs when you subtract a binomial!

#examplebox("Example 5")[][
Subtract: #math.equation(block: false, alt: "the fraction y squared over y minus 6 minus the fraction 2 y plus 24 over y minus 6 .")[$frac(y^(2), y − 6) − frac(2 y + 24, y − 6) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction y squared over y minus 6 minus the fraction 2 y plus 24 over y minus 6")[$frac(y^(2), y − 6) − frac(2 y + 24, y − 6)$]]),
  [The fractions have a common #linebreak() denominator, so subtract the numerators #linebreak() and place the difference over the common denominator.], [#math.equation(block: false, alt: "the fraction y squared minus open parenthesis 2 y plus 24 close parenthesis over y minus 6")[$frac(y^(2) − ( 2 y + 24 ), y − 6)$]],
  [Distribute the sign in the numerator.], [#math.equation(block: false, alt: "the fraction y squared minus 2 y minus 24 over y minus 6")[$frac(y^(2) − 2 y − 24, y − 6)$]],
  [Factor the numerator.], [#math.equation(block: false, alt: "the fraction open parenthesis y minus 6 close parenthesis open parenthesis y plus 4 close parenthesis over y minus 6")[$frac(( y − 6 ) ( y + 4 ), y − 6)$]],
  [Remove common factors.], [#math.equation(block: false, alt: "the fraction open parenthesis y minus 6 close parenthesis open parenthesis y plus 4 close parenthesis over y minus 6")[$frac(cancel(( y − 6 )) ( y + 4 ), cancel(y − 6))$]],
  [Simplify.], [#math.equation(block: false, alt: "y plus 4")[$y + 4$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Subtract: #math.equation(block: false, alt: "the fraction n squared over n minus 4 minus the fraction n plus 12 over n minus 4 .")[$frac(n^(2), n − 4) − frac(n + 12, n − 4) .$]

#solutionbox[
#math.equation(block: true, alt: "n plus 3")[$n + 3$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Subtract: #math.equation(block: false, alt: "the fraction y squared over y minus 1 minus the fraction 9 y minus 8 over y minus 1 .")[$frac(y^(2), y − 1) − frac(9 y − 8, y − 1) .$]

#solutionbox[
#math.equation(block: true, alt: "y minus 8")[$y − 8$]
]
]

#examplebox("Example 6")[][
Subtract: #math.equation(block: false, alt: "the fraction 5 x squared minus 7 x plus 3 over x squared minus 3 x minus 18 minus the fraction 4 x squared plus x minus 9 over x squared minus 3 x minus 18 .")[$frac(5 x^(2) − 7 x + 3, x^(2) − 3 x − 18) − frac(4 x^(2) + x − 9, x^(2) − 3 x − 18) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 5 x squared minus 7 x plus 3 over x squared minus 3 x minus 18 minus the fraction 4 x squared plus x minus 9 over x squared minus 3 x minus 18")[$frac(5 x^(2) − 7 x + 3, x^(2) − 3 x − 18) − frac(4 x^(2) + x − 9, x^(2) − 3 x − 18)$]]),
  [Subtract the numerators and place the #linebreak() difference over the common denominator.], [#math.equation(block: false, alt: "the fraction 5 x squared minus 7 x plus 3 minus open parenthesis 4 x squared plus x minus 9 close parenthesis over x squared minus 3 x minus 18")[$frac(5 x^(2) − 7 x + 3 − ( 4 x^(2) + x − 9 ), x^(2) − 3 x − 18)$]],
  [Distribute the sign in the numerator.], [#math.equation(block: false, alt: "the fraction 5 x squared minus 7 x plus 3 minus 4 x squared minus x plus 9 over x squared minus 3 x minus 18")[$frac(5 x^(2) − 7 x + 3 − 4 x^(2) − x + 9, x^(2) − 3 x − 18)$]],
  [Combine like terms.], [#math.equation(block: false, alt: "the fraction x squared minus 8 x plus 12 over x squared minus 3 x minus 18")[$frac(x^(2) − 8 x + 12, x^(2) − 3 x − 18)$]],
  [Factor the numerator and the denominator.], [#math.equation(block: false, alt: "the fraction open parenthesis x minus 2 close parenthesis open parenthesis x minus 6 close parenthesis over open parenthesis x plus 3 close parenthesis open parenthesis x minus 6 close parenthesis")[$frac(( x − 2 ) ( x − 6 ), ( x + 3 ) ( x − 6 ))$]],
  [Simplify by removing common factors.], [#math.equation(block: false, alt: "the fraction open parenthesis x minus 2 close parenthesis open parenthesis x minus 6 close parenthesis over open parenthesis x plus 3 close parenthesis open parenthesis x minus 6 close parenthesis")[$frac(( x − 2 ) cancel(( x − 6 )), ( x + 3 ) cancel(( x − 6 )))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction open parenthesis x minus 2 close parenthesis over open parenthesis x plus 3 close parenthesis")[$frac(( x − 2 ), ( x + 3 ))$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Subtract: #math.equation(block: false, alt: "the fraction 4 x squared minus 11 x plus 8 over x squared minus 3 x plus 2 minus the fraction 3 x squared plus x minus 3 over x squared minus 3 x plus 2 .")[$frac(4 x^(2) − 11 x + 8, x^(2) − 3 x + 2) − frac(3 x^(2) + x − 3, x^(2) − 3 x + 2) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction x minus 11 over x minus 2")[$frac(x − 11, x − 2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Subtract: #math.equation(block: false, alt: "the fraction 6 x squared minus x plus 20 over x squared minus 81 minus the fraction 5 x squared plus 11 x minus 7 over x squared minus 81 .")[$frac(6 x^(2) − x + 20, x^(2) − 81) − frac(5 x^(2) + 11 x − 7, x^(2) − 81) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction x minus 3 over x plus 9")[$frac(x − 3, x + 9)$]
]
]


=== Add and Subtract Rational Expressions whose Denominators are Opposites

When the denominators of two rational expressions are opposites, it is easy to get a common denominator. We just have to multiply one of the fractions by #math.equation(block: false, alt: "the fraction −1 over −1")[$frac(−1, −1)$].

Let’s see how this works.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression showing the sum of two fractions: 7 over d plus 5 over negative d.], alt: "A mathematical expression showing the sum of two fractions: 7 over d plus 5 over negative d.", caption: none)]),
  [Multiply the second fraction by #math.equation(block: false, alt: "the fraction −1 over −1")[$frac(−1, −1)$].], [#figure(figph[An algebraic expression showing the sum of two fractions. The first term is 7/d, and the second term is a fraction with a numerator of (-1) to the power of 5 and a denominator of (-1)(-d).], alt: "An algebraic expression showing the sum of two fractions. The first term is 7/d, and the second term is a fraction with a numerator of (-1) to the power of 5 and a denominator of (-1)(-d).", caption: none)],
  [The denominators are the same.], [#figure(figph[The image shows the mathematical expression 7/d + -5/d, representing the addition of two fractions with a common denominator 'd'.], alt: "The image shows the mathematical expression 7/d + -5/d, representing the addition of two fractions with a common denominator 'd'.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression displaying the fraction 2 over d.], alt: "A mathematical expression displaying the fraction 2 over d.", caption: none)],
))

#examplebox("Example 7")[][
Add: #math.equation(block: false, alt: "the fraction 4 u minus 1 over 3 u minus 1 plus the fraction u over 1 minus 3 u .")[$frac(4 u − 1, 3 u − 1) + frac(u, 1 − 3 u) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[Algebraic expression: (4u-1)/(3u-1) + u/(1-3u). It's a sum of two fractions with denominators that are additive inverses of each other.], alt: "Algebraic expression: (4u-1)/(3u-1) + u/(1-3u). It's a sum of two fractions with denominators that are additive inverses of each other.", caption: none)]),
  [The denominators are opposites, so multiply the second fraction by #math.equation(block: false, alt: "the fraction −1 over −1")[$frac(−1, −1)$].], [#figure(figph[A mathematical expression representing the sum of two algebraic fractions. The first term is (4u-1)/(3u-1), and the second is ((-1)u)/((-1)(1-3u)), illustrating a step in simplifying the expression.], alt: "A mathematical expression representing the sum of two algebraic fractions. The first term is (4u-1)/(3u-1), and the second is ((-1)u)/((-1)(1-3u)), illustrating a step in simplifying the expression.", caption: none)],
  [Simplify the second fraction.], [#figure(figph[The image displays the sum of two algebraic fractions with a common denominator, written as (4u-1)/(3u-1) + (-u)/(3u-1).], alt: "The image displays the sum of two algebraic fractions with a common denominator, written as (4u-1)/(3u-1) + (-u)/(3u-1).", caption: none)],
  [The denominators are the same. Add the numerators.], [#figure(figph[A mathematical fraction displays '4u - 1 - u' as the numerator and '3u - 1' as the denominator, which simplifies to (3u - 1) / (3u - 1).], alt: "A mathematical fraction displays '4u - 1 - u' as the numerator and '3u - 1' as the denominator, which simplifies to (3u - 1) / (3u - 1).", caption: none)],
  [Simplify.], [#figure(figph[The mathematical expression showing the fraction (3u-1) divided by (3u-1). This expression simplifies to 1, assuming that (3u-1) is not equal to zero.], alt: "The mathematical expression showing the fraction (3u-1) divided by (3u-1). This expression simplifies to 1, assuming that (3u-1) is not equal to zero.", caption: none)],
  [Simplify.], [#figure(figph[The number 1.], alt: "The number 1.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Add: #math.equation(block: false, alt: "the fraction 8 x minus 15 over 2 x minus 5 plus the fraction 2 x over 5 minus 2 x .")[$frac(8 x − 15, 2 x − 5) + frac(2 x, 5 − 2 x) .$]

#solutionbox[
#math.equation(block: true, alt: "3")[$3$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Add: #math.equation(block: false, alt: "the fraction 6 y squared plus 7 y minus 10 over 4 y minus 7 plus the fraction 2 y squared plus 2 y plus 11 over 7 minus 4 y .")[$frac(6 y^(2) + 7 y − 10, 4 y − 7) + frac(2 y^(2) + 2 y + 11, 7 − 4 y) .$]

#solutionbox[
#math.equation(block: true, alt: "y plus 3")[$y + 3$]
]
]

#examplebox("Example 8")[][
Subtract: #math.equation(block: false, alt: "the fraction m squared minus 6 m over m squared minus 1 minus the fraction 3 m plus 2 over 1 minus m squared .")[$frac(m^(2) − 6 m, m^(2) − 1) − frac(3 m + 2, 1 − m^(2)) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical expression showing the subtraction of two algebraic fractions: (m^2 - 6m) / (m^2 - 1) - (3m + 2) / (1 - m^2).], alt: "A mathematical expression showing the subtraction of two algebraic fractions: (m^2 - 6m) / (m^2 - 1) - (3m + 2) / (1 - m^2).", caption: none)]),
  [The denominators are opposites, so multiply the second fraction by #math.equation(block: false, alt: "the fraction −1 over −1")[$frac(−1, −1)$].], [#figure(figph[A mathematical expression displaying the subtraction of two algebraic fractions. The first fraction is (m^2 - 6m)/(m^2 - 1), and the second is -1(3m + 2)/-1(1 - m^2).], alt: "A mathematical expression displaying the subtraction of two algebraic fractions. The first fraction is (m^2 - 6m)/(m^2 - 1), and the second is -1(3m + 2)/-1(1 - m^2).", caption: none)],
  [Simplify the second fraction.], [#figure(figph[A mathematical expression showing the subtraction of two algebraic fractions with a common denominator of m^2 - 1. The first fraction is (m^2 - 6m) / (m^2 - 1) and the second is (-3m - 2) / (m^2 - 1).], alt: "A mathematical expression showing the subtraction of two algebraic fractions with a common denominator of m^2 - 1. The first fraction is (m^2 - 6m) / (m^2 - 1) and the second is (-3m - 2) / (m^2 - 1).", caption: none)],
  [The denominators are the same. Subtract the numerators.], [#figure(figph[A mathematical expression showing the fraction (m^2 - 6m - (-3m - 2)) / (m^2 - 1).], alt: "A mathematical expression showing the fraction (m^2 - 6m - (-3m - 2)) / (m^2 - 1).", caption: none)],
  [Distribute.], [#math.equation(block: false, alt: "the fraction m squared minus 6 m plus 3 m plus 2 over m squared minus 1")[$frac(m^(2) − 6 m + 3 m + 2, m^(2) − 1)$]],
  [Combine like terms.], [#figure(figph[A mathematical expression showing a fraction with the numerator m squared minus 3m plus 2, and the denominator m squared minus 1.], alt: "A mathematical expression showing a fraction with the numerator m squared minus 3m plus 2, and the denominator m squared minus 1.", caption: none)],
  [Factor the numerator and denominator.], [#figure(figph[A mathematical fraction displaying (m-1)(m-2) in the numerator and (m-1)(m+1) in the denominator, illustrating an algebraic expression suitable for simplification.], alt: "A mathematical fraction displaying (m-1)(m-2) in the numerator and (m-1)(m+1) in the denominator, illustrating an algebraic expression suitable for simplification.", caption: none)],
  [Simplify by removing common factors.], [#figure(figph[A fraction showing the expression (m-1)(m-2) divided by (m-1)(m+1), with the (m-1) terms crossed out in both the numerator and denominator, implying simplification to (m-2)/(m+1).], alt: "A fraction showing the expression (m-1)(m-2) divided by (m-1)(m+1), with the (m-1) terms crossed out in both the numerator and denominator, implying simplification to (m-2)/(m+1).", caption: none)],
  [Simplify.], [#figure(figph[A mathematical fraction displays 'm minus 2' in the numerator and 'm plus 1' in the denominator, set against a plain white background.], alt: "A mathematical fraction displays 'm minus 2' in the numerator and 'm plus 1' in the denominator, set against a plain white background.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Subtract: #math.equation(block: false, alt: "the fraction y squared minus 5 y over y squared minus 4 minus the fraction 6 y minus 6 over 4 minus y squared .")[$frac(y^(2) − 5 y, y^(2) − 4) − frac(6 y − 6, 4 − y^(2)) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Subtract: #math.equation(block: false, alt: "the fraction 2 n squared plus 8 n minus 1 over n squared minus 1 minus the fraction n squared minus 7 n minus 1 over 1 minus n squared .")[$frac(2 n^(2) + 8 n − 1, n^(2) − 1) − frac(n^(2) − 7 n − 1, 1 − n^(2)) .$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 n minus 2 over n minus 1")[$frac(3 n − 2, n − 1)$]
]
]


=== Key Concepts

- #strong[Rational Expression Addition]
- If #math.equation(block: false, alt: "p , q , and r")[$p , q , #h(0.2em) "and" #h(0.2em) r$] are polynomials where #math.equation(block: false, alt: "r not equal to 0")[$r ≠ 0$], then #linebreak() 
#math.equation(block: true, alt: "the fraction p over r plus the fraction q over r equals the fraction p plus q over r")[$frac(p, r) + frac(q, r) = frac(p + q, r)$]
- To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator.
- #strong[Rational Expression Subtraction]
- If #math.equation(block: false, alt: "p , q , and r")[$p , q , #h(0.2em) "and" #h(0.2em) r$] are polynomials where #math.equation(block: false, alt: "r not equal to 0")[$r ≠ 0$], then #linebreak() 
#math.equation(block: true, alt: "the fraction p over r minus the fraction q over r equals the fraction p minus q over r")[$frac(p, r) − frac(q, r) = frac(p − q, r)$]
- To subtract rational expressions, subtract the numerators and place the difference over the common denominator.


==== Practice Makes Perfect

#strong[Add Rational Expressions with a Common Denominator]

In the following exercises, add.

#math.equation(block: true, alt: "the fraction 2 over 15 plus the fraction 7 over 15")[$frac(2, 15) + frac(7, 15)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 over 5")[$frac(3, 5)$]
]

#math.equation(block: true, alt: "the fraction 4 over 21 plus the fraction 3 over 21")[$frac(4, 21) + frac(3, 21)$]

#math.equation(block: true, alt: "the fraction 7 over 24 plus the fraction 11 over 24")[$frac(7, 24) + frac(11, 24)$]

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

#math.equation(block: true, alt: "the fraction 7 over 36 plus the fraction 13 over 36")[$frac(7, 36) + frac(13, 36)$]

#math.equation(block: true, alt: "the fraction 3 a over a minus b plus the fraction 1 over a minus b")[$frac(3 a, a − b) + frac(1, a − b)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 a plus 1 over a minus b")[$frac(3 a + 1, a − b)$]
]

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

#math.equation(block: true, alt: "the fraction d over d plus 8 plus the fraction 5 over d plus 8")[$frac(d, d + 8) + frac(5, d + 8)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction d plus 5 over d plus 8")[$frac(d + 5, d + 8)$]
]

#math.equation(block: true, alt: "the fraction 7 m over 2 m plus n plus the fraction 4 over 2 m plus n")[$frac(7 m, 2 m + n) + frac(4, 2 m + n)$]

#math.equation(block: true, alt: "the fraction p squared plus 10 p over p plus 2 plus the fraction 16 over p plus 2")[$frac(p^(2) + 10 p, p + 2) + frac(16, p + 2)$]

#solutionbox[
#math.equation(block: true, alt: "p plus 8")[$p + 8$]
]

#math.equation(block: true, alt: "the fraction q squared plus 12 q over q plus 3 plus the fraction 27 over q plus 3")[$frac(q^(2) + 12 q, q + 3) + frac(27, q + 3)$]

#math.equation(block: true, alt: "the fraction 2 r squared over 2 r minus 1 plus the fraction 15 r minus 8 over 2 r minus 1")[$frac(2 r^(2), 2 r − 1) + frac(15 r − 8, 2 r − 1)$]

#solutionbox[
#math.equation(block: true, alt: "r plus 8")[$r + 8$]
]

#math.equation(block: true, alt: "the fraction 3 s squared over 3 s minus 2 plus the fraction 13 s minus 10 over 3 s minus 2")[$frac(3 s^(2), 3 s − 2) + frac(13 s − 10, 3 s − 2)$]

#math.equation(block: true, alt: "the fraction 8 t squared over t plus 4 plus the fraction 32 t over t plus 4")[$frac(8 t^(2), t + 4) + frac(32 t, t + 4)$]

#solutionbox[
#math.equation(block: true, alt: "8 t")[$8 t$]
]

#math.equation(block: true, alt: "the fraction 6 v squared over v plus 5 plus the fraction 30 v over v plus 5")[$frac(6 v^(2), v + 5) + frac(30 v, v + 5)$]

#math.equation(block: true, alt: "the fraction 2 w squared over w squared minus 16 plus the fraction 8 w over w squared minus 16")[$frac(2 w^(2), w^(2) − 16) + frac(8 w, w^(2) − 16)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 w over w minus 4")[$frac(2 w, w − 4)$]
]

#math.equation(block: true, alt: "the fraction 7 x squared over x squared minus 9 plus the fraction 21 x over x squared minus 9")[$frac(7 x^(2), x^(2) − 9) + frac(21 x, x^(2) − 9)$]

#strong[Subtract Rational Expressions with a Common Denominator]

In the following exercises, subtract.

#math.equation(block: true, alt: "the fraction y squared over y plus 8 minus the fraction 64 over y plus 8")[$frac(y^(2), y + 8) − frac(64, y + 8)$]

#solutionbox[
#math.equation(block: true, alt: "y minus 8")[$y − 8$]
]

#math.equation(block: true, alt: "the fraction z squared over z plus 2 minus the fraction 4 over z plus 2")[$frac(z^(2), z + 2) − frac(4, z + 2)$]

#math.equation(block: true, alt: "the fraction 9 a squared over 3 a minus 7 minus the fraction 49 over 3 a minus 7")[$frac(9 a^(2), 3 a − 7) − frac(49, 3 a − 7)$]

#solutionbox[
#math.equation(block: true, alt: "3 a plus 7")[$3 a + 7$]
]

#math.equation(block: true, alt: "the fraction 25 b squared over 5 b minus 6 minus the fraction 36 over 5 b minus 6")[$frac(25 b^(2), 5 b − 6) − frac(36, 5 b − 6)$]

#math.equation(block: true, alt: "the fraction c squared over c minus 8 minus the fraction 6 c plus 16 over c minus 8")[$frac(c^(2), c − 8) − frac(6 c + 16, c − 8)$]

#solutionbox[
#math.equation(block: true, alt: "c plus 2")[$c + 2$]
]

#math.equation(block: true, alt: "the fraction d squared over d minus 9 minus the fraction 6 d plus 27 over d minus 9")[$frac(d^(2), d − 9) − frac(6 d + 27, d − 9)$]

#math.equation(block: true, alt: "the fraction 3 m squared over 6 m minus 30 minus the fraction 21 m minus 30 over 6 m minus 30")[$frac(3 m^(2), 6 m − 30) − frac(21 m − 30, 6 m − 30)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction m minus 2 over 2")[$frac(m − 2, 2)$]
]

#math.equation(block: true, alt: "the fraction 2 n squared over 4 n minus 32 minus the fraction 18 n minus 16 over 4 n minus 32")[$frac(2 n^(2), 4 n − 32) − frac(18 n − 16, 4 n − 32)$]

#math.equation(block: true, alt: "the fraction 6 p squared plus 3 p plus 4 over p squared plus 4 p minus 5 minus the fraction 5 p squared plus p plus 7 over p squared plus 4 p minus 5")[$frac(6 p^(2) + 3 p + 4, p^(2) + 4 p − 5) − frac(5 p^(2) + p + 7, p^(2) + 4 p − 5)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction p plus 3 over p plus 5")[$frac(p + 3, p + 5)$]
]

#math.equation(block: true, alt: "the fraction 5 q squared plus 3 q minus 9 over q squared plus 6 q plus 8 minus the fraction 4 q squared plus 9 q plus 7 over q squared plus 6 q plus 8")[$frac(5 q^(2) + 3 q − 9, q^(2) + 6 q + 8) − frac(4 q^(2) + 9 q + 7, q^(2) + 6 q + 8)$]

#math.equation(block: true, alt: "the fraction 5 r squared plus 7 r minus 33 over r squared minus 49 minus the fraction 4 r squared plus 5 r plus 30 over r squared minus 49")[$frac(5 r^(2) + 7 r − 33, r^(2) − 49) − frac(4 r^(2) + 5 r + 30, r^(2) − 49)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction r plus 9 over r plus 7")[$frac(r + 9, r + 7)$]
]

#math.equation(block: true, alt: "the fraction 7 t squared minus t minus 4 over t squared minus 25 minus the fraction 6 t squared plus 12 t minus 44 over t squared minus 25")[$frac(7 t^(2) − t − 4, t^(2) − 25) − frac(6 t^(2) + 12 t − 44, t^(2) − 25)$]

#strong[Add and Subtract Rational Expressions whose Denominators are Opposites]

In the following exercises, add.

#math.equation(block: true, alt: "the fraction 10 v over 2 v minus 1 plus the fraction 2 v plus 4 over 1 minus 2 v")[$frac(10 v, 2 v − 1) + frac(2 v + 4, 1 − 2 v)$]

#solutionbox[
#math.equation(block: true, alt: "4")[$4$]
]

#math.equation(block: true, alt: "the fraction 20 w over 5 w minus 2 plus the fraction 5 w plus 6 over 2 minus 5 w")[$frac(20 w, 5 w − 2) + frac(5 w + 6, 2 − 5 w)$]

#math.equation(block: true, alt: "the fraction 10 x squared plus 16 x minus 7 over 8 x minus 3 plus the fraction 2 x squared plus 3 x minus 1 over 3 minus 8 x")[$frac(10 x^(2) + 16 x − 7, 8 x − 3) + frac(2 x^(2) + 3 x − 1, 3 − 8 x)$]

#solutionbox[
#math.equation(block: true, alt: "x plus 2")[$x + 2$]
]

#math.equation(block: true, alt: "the fraction 6 y squared plus 2 y minus 11 over 3 y minus 7 plus the fraction 3 y squared minus 3 y plus 17 over 7 minus 3 y")[$frac(6 y^(2) + 2 y − 11, 3 y − 7) + frac(3 y^(2) − 3 y + 17, 7 − 3 y)$]

In the following exercises, subtract.

#math.equation(block: true, alt: "the fraction z squared plus 6 z over z squared minus 25 minus the fraction 3 z plus 20 over 25 minus z squared")[$frac(z^(2) + 6 z, z^(2) − 25) − frac(3 z + 20, 25 − z^(2))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction z plus 4 over z minus 5")[$frac(z + 4, z − 5)$]
]

#math.equation(block: true, alt: "the fraction a squared plus 3 a over a squared minus 9 minus the fraction 3 a minus 27 over 9 minus a squared")[$frac(a^(2) + 3 a, a^(2) − 9) − frac(3 a − 27, 9 − a^(2))$]

#math.equation(block: true, alt: "the fraction 2 b squared plus 30 b minus 13 over b squared minus 49 minus the fraction 2 b squared minus 5 b minus 8 over 49 minus b squared")[$frac(2 b^(2) + 30 b − 13, b^(2) − 49) − frac(2 b^(2) − 5 b − 8, 49 − b^(2))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 b minus 3 over b minus 7")[$frac(4 b − 3, b − 7)$]
]

#math.equation(block: true, alt: "the fraction c squared plus 5 c minus 10 over c squared minus 16 minus the fraction c squared minus 8 c minus 10 over 16 minus c squared")[$frac(c^(2) + 5 c − 10, c^(2) − 16) − frac(c^(2) − 8 c − 10, 16 − c^(2))$]


==== Everyday Math

Sarah ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. If #math.equation(block: false, alt: "r")[$r$] represents Sarah’s speed when she ran, then her running time is modeled by the expression #math.equation(block: false, alt: "the fraction 8 over r")[$frac(8, r)$] and her biking time is modeled by the expression #math.equation(block: false, alt: "the fraction 24 over r plus 4 .")[$frac(24, r + 4) .$] Add the rational expressions #math.equation(block: false, alt: "the fraction 8 over r plus the fraction 24 over r plus 4")[$frac(8, r) + frac(24, r + 4)$] to get an expression for the total amount of time Sarah ran and biked.

#solutionbox[
#math.equation(block: true, alt: "the fraction 32 open parenthesis r plus 1 close parenthesis over r open parenthesis r plus 4 close parenthesis")[$frac(32 ( r + 1 ), r ( r + 4 ))$]
]

If Pete can paint a wall in #math.equation(block: false, alt: "p")[$p$] hours, then in one hour he can paint #math.equation(block: false, alt: "the fraction 1 over p")[$frac(1, p)$] of the wall. It would take Penelope 3 hours longer than Pete to paint the wall, so in one hour she can paint #math.equation(block: false, alt: "the fraction 1 over p plus 3")[$frac(1, p + 3)$] of the wall. Add the rational expressions #math.equation(block: false, alt: "the fraction 1 over p plus the fraction 1 over p plus 3")[$frac(1, p) + frac(1, p + 3)$] to get an expression for the part of the wall Pete and Penelope would paint in one hour if they worked together.


==== Writing Exercises

Donald thinks that #math.equation(block: false, alt: "the fraction 3 over x plus the fraction 4 over x")[$frac(3, x) + frac(4, x)$] is #math.equation(block: false, alt: "the fraction 7 over 2 x .")[$frac(7, 2 x) .$] Is Donald correct? Explain.

Explain how you find the Least Common Denominator of #math.equation(block: false, alt: "x squared plus 5 x plus 4")[$x^(2) + 5 x + 4$] and #math.equation(block: false, alt: "x squared minus 16 .")[$x^(2) − 16 .$]


==== Self Check

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

#figure(figph[The above image is a table with four columns and four rows. The first row is the header row. The first header is labeled “I can…”, the second “Confidently”, the third, “With some help”, and the fourth “No – I don’t get it!”. In the first column under “I can”, the next row reads “add rational expressions with a common denominator.”, the next row reads “subtract rational expressions with a common denominator.”, the next row reads, “add and subtract rational expressions whose denominators are opposites.”, the last row reads “What does this checklist tell you about your mastery of this section? What steps will you take to improve?” The remaining columns are blank.], alt: "The above image is a table with four columns and four rows. The first row is the header row. The first header is labeled “I can…”, the second “Confidently”, the third, “With some help”, and the fourth “No – I don’t get it!”. In the first column under “I can”, the next row reads “add rational expressions with a common denominator.”, the next row reads “subtract rational expressions with a common denominator.”, the next row reads, “add and subtract rational expressions whose denominators are opposites.”, the last row reads “What does this checklist tell you about your mastery of this section? What steps will you take to improve?” The remaining columns are blank.", caption: none)

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?