Free Printable WorksheetMathPre-CalculusMath CurriculumPre-CAL.51

Pre CAL.51 Solve Rational Inequalities Using Sign Charts Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Pre-Calculus Math worksheet provides structured practice on Pre CAL.51 Solve Rational Inequalities Using Sign Charts. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

Pre-CAL.51 Solve Rational Inequalities Using Sign Charts - Solve Rational Inequalities Using Sign Charts

Whether you are an educator preparing classroom handouts, a tutor reinforcing weekly topics, or a parent supporting homeschool learning, this printable resource gives students the deliberate practice required to achieve subject fluency and master state standards.

Grade

Pre-Calculus

Subject

Math

Domain

Math Curriculum

Formats

PDF & DOCX

Sample Practice Problems

Preview the types of problems and questions covered in this worksheet

Below are sample questions representative of the exercises included on this worksheet and related Pre CAL.51 Solve Rational Inequalities Using Sign Charts skill assessments:

Problem 1Difficulty: Mastery

Solve the rational inequality using a sign chart. (Exclude values that make the denominator 0.) -(x + 8)(x + 8)(x - 5)(x - 6)(x - 6) ÷ (x + 3)(x - 4)(x - 7)(x - 7) <= 0

A(-infinity, -8) U (-8, -3) U (4, 5)

B(-infinity, -3) U (4, 5] U {6}

C[-3, 4] U [5, 7] U [7, infinity) U {-8}

D(-3, 4) U [5, 7) U (7, infinity) U {-8}

Solution & Explanation:Find critical points: zeros of the numerator (x = -8, x = 5, x = 6) and zeros of the denominator (x = -3, x = 4, x = 7). These critical points split the number line into intervals. Note: the factor (x - 6) repeats 2 times in the numerator, so the sign does not change at x = 6 if m is even; the factor (x + 8) repeats 2 times in the numerator, so the sign does not change at x = -8 if m is even; the factor (x - 7) repeats 2 times in the denominator, so the sign does not change at x = 7 if m is even (but x = 7 is still excluded). There is a negative sign out front, so the sign of the rational expression is flipped on every interval. Test one value in each interval to determine whether the expression is positive or negative, then keep the interval(s) where the expression is <= 0. Include numerator zeros only when the inequality allows equality (≥ or ≤), and never include denominator zeros. Therefore, the solution set is (-3, 4) U [5, 7) U (7, infinity) U {-8}.

Problem 2Difficulty: Mastery

Solve the rational inequality using a sign chart. (Exclude values that make the denominator 0.) (x + 7)(x + 7)(x + 7)(x + 1)(x - 6)(x - 6) ÷ (x + 3)(x + 2)(x - 4) < 0

A(-7, -3) U (-2, -1) U (4, 6) U (6, infinity)

B[-7, -3) U (-2, -1] U (4, infinity)

C(-infinity, -7) U (-3, -2) U (-1, 4)

D(-infinity, -7] U (-3, -2) U [-1, 4) U {6}

Solution & Explanation:Find critical points: zeros of the numerator (x = -7, x = -1, x = 6) and zeros of the denominator (x = -3, x = -2, x = 4). These critical points split the number line into intervals. Note: the factor (x + 7) repeats 3 times in the numerator, so the sign does not change at x = -7 if m is even; the factor (x - 6) repeats 2 times in the numerator, so the sign does not change at x = 6 if m is even. Test one value in each interval to determine whether the expression is positive or negative, then keep the interval(s) where the expression is < 0. Include numerator zeros only when the inequality allows equality (≥ or ≤), and never include denominator zeros. Therefore, the solution set is (-infinity, -7) U (-3, -2) U (-1, 4).

Problem 3Difficulty: Mastery

Solve the rational inequality using a sign chart. (Exclude values that make the denominator 0.) (x + 6)(x + 3)(x + 2)(x - 8) ÷ (x - 6)(x - 7) >= 0

A[-6, -3] U [-2, 6) U (7, 8]

B(-infinity, -6] U [-3, -2] U (6, 7) U [8, infinity)

C(-6, -3) U (-2, 6) U (7, 8)

D(-infinity, -6) U (-3, -2) U (6, 7) U (8, infinity)

Solution & Explanation:Find critical points: zeros of the numerator (x = -6, x = -3, x = -2, x = 8) and zeros of the denominator (x = 6, x = 7). These critical points split the number line into intervals. Test one value in each interval to determine whether the expression is positive or negative, then keep the interval(s) where the expression is >= 0. Include numerator zeros only when the inequality allows equality (≥ or ≤), and never include denominator zeros. Therefore, the solution set is (-infinity, -6] U [-3, -2] U (6, 7) U [8, infinity).

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QWhat grade level and standards does this worksheet align with?

This worksheet is designed specifically for Pre-Calculus Math students. It aligns with educational standard Pre-CAL.51 (Solve Rational Inequalities Using Sign Charts).