A.10(D)·Algebra II·Rational FunctionsSupporting

Solving Rational Equations

Solve rational equations and verify the resulting value.

Students solve rational equations by noting excluded values, clearing denominators, solving the resulting equation, and verifying the value in the original equation.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Algebra II key. Algebra II is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutor All Algebra II released practice45 minutes

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Excluded values

A rational equation has a variable in a denominator. Values that make a denominator zero are excluded. In $\frac{18}{3x-2}=6$ , $x\ne\frac{2}{3}$ .

Clear the denominators

Multiply both sides by the denominator or least common denominator. Distribute carefully: $6(3x-2)=18x-12$ . Then solve the resulting linear or quadratic equation.

Check and record

Substitute the result into the original equation. Reject any excluded value. For a griddable answer, convert fractions to an exact decimal when possible, such as $\frac{6}{5}=1.2$ .

Worked example

Solve $\frac{18}{3x-2}=6$ .

Excluded value: $x\ne\frac{2}{3}$ .
Multiply both sides by $3x-2$ : $18=6(3x-2)$ .
Distribute and solve: $18=18x-12$ , so $30=18x$ and $x=\frac{5}{3}$ .
Check: $3\left(\frac{5}{3}\right)-2=3$ , and $\frac{18}{3}=6$ .

$x=\frac{5}{3}$

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Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Algebra 2 Solving Contextual Rational Equations

UT IPSI · Kid2Kid / Texas Education Agency

Solve rational equations, check the solutions, and interpret them in context.

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Try it yourself

TEKSForge practice · unscored

1. Solve $\frac{x}{x-3}=4$ .
Show answer and reasoning
$x=4$
Multiply by $x-3$ : $x=4x-12$ , so $x=4$ . The value is not excluded, and $\frac{4}{1}=4$ .
2. Solve $\frac{2}{x}+\frac{1}{2}=1$ .
Show answer and reasoning
$x=4$
Multiply every term by $2x$ : $4+x=2x$ , so $x=4$ . Check: $\frac{2}{4}+\frac{1}{2}=1$ .

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Algebra II EOC · 2013 · Item 13 · 1 point

What value of $p$ makes the equation $\frac{19}{4p - 1} = 5$ true? Record your answer.

Show STAAR answer and explanation

1.2

Multiply by $4p-1$ to get $19=5(4p-1)=20p-5$ . Add 5 and divide by 20: $p=24/20=1.2$ .

1: Substituting 1 gives $19/3$ , not 5. Solve $19=20p-5$ to get $p=1.2$ .

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Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

Identify values that make a denominator zero.
Multiply both sides by a denominator or least common denominator to clear fractions.
Verify solutions and record exact or decimal answers accurately.

Instructional sequence

1
Engage
5 min
Ask what value of x would make $\frac{12}{x}=3$ true and how students know.
2
Direct instruction
10 min
Model listing excluded values, multiplying both sides by the denominator, and solving.
3
Guided practice
12 min
Solve one-denominator equations, then equations with two fractions using the least common denominator.
4
Graphical check
12 min
Graph each side as a function and confirm the intersection matches the algebraic solution.
5
Closure
6 min
Complete the exit ticket and write the answer as it would be gridded.

Misconceptions to address

““Multiply only the fraction by its denominator.””

Teacher move: To keep the equation balanced, multiply both entire sides by the same expression.

““A value that makes a denominator zero can still be a solution.””

Teacher move: Division by zero is undefined. Excluded values must be rejected even if they appear during solving.

Materials

•Rational-equation organizer
•Graphing calculator
•Griddable answer practice sheet

Teaching tips

Write excluded values beside the equation before solving.
Use parentheses when multiplying by a binomial denominator.
Convert fractional answers to decimals carefully for griddable responses.

Exit ticket · 3–5 min

Solve $\frac{10}{2x+1}=2$ .

Show expected answer

$10=2(2x+1)$ , so $10=4x+2$ and $x=2$ . Check: $\frac{10}{5}=2$ .

Sources & alignment

TEA 2013 Algebra II released test TEA 2013 Algebra II answer key and standards Texas Gateway, home of the UT IPSI Kid2Kid lessons

Alignment follows the standard TEA assigned to each 2013 released item. Video playback depends on YouTube access; written instruction remains available here.

Learn the concept

Excluded values

Clear the denominators

Check and record

Worked example

Watch the lesson

Algebra 2 Solving Contextual Rational Equations

Try it yourself

1. Solve xx−3=4\frac{x}{x-3}=4x−3x​=4.

2. Solve 2x+12=1\frac{2}{x}+\frac{1}{2}=1x2​+21​=1.

Released STAAR practice

Archived released STAAR Algebra II EOC · 2013 · Item 13 · 1 point

Learning objectives

Instructional sequence

Engage

Direct instruction

Guided practice

Graphical check

Closure

Misconceptions to address

1. Solve $\frac{x}{x-3}=4$ .

2. Solve $\frac{2}{x}+\frac{1}{2}=1$ .