MathLab with Mr. Ahmed

MathLab with Mr. Ahmed is a math learning website by Ahmed Amasha, a mathematics teacher teaching the American curriculum. Explore Algebra 2, Precalculus, Calculus, and SAT and PSAT Math Prep through lessons, focused practice and printable resources.

MathLabwith Mr. Ahmed
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1.8 Inequalities

Solve and write solution intervals.

Before you solve

Ideas in this lesson

Review the ideas below. Select one to see worked steps and five practice questions.

01Test solutions and endpoint restrictionsWorked steps · five questions02Solve linear inequalitiesWorked steps · five questions03Combine compound inequalitiesWorked steps · five questions04Solve polynomial inequalities with sign chartsWorked steps · five questions05Solve rational inequalities and preserve exclusionsWorked steps · five questions06Solve absolute-value inequalitiesWorked steps · five questions07Translate distance and tolerance descriptionsWorked steps · five questions08Use inequalities in domains and modelsWorked steps · five questions09Solve parameter-dependent inequalitiesWorked steps · five questions10Translate and solve unit-sensitive linear modelsWorked steps · five questions11Analyze nonlinear physical and economic modelsWorked steps · five questions12Prove order properties and compare distancesWorked steps · five questions
Lesson brief and rules

1.8 Inequalities

Apply the same operation to both sides; reverse the inequality sign when multiplying or dividing by a negative number.

Remember
An open endpoint is used for < or >.
Worked example 11
01 / Which listed values {1,2,3,4}\{1,2,3,4\} satisfy 5x−2(x−1)≤85x-2(x-1)\le 8?
02 / Simplify the left side: 5x−2(x−1)=3x+25x-2(x-1)=3x+2.
03 / The inequality becomes 3x+2≤83x+2\le 8, hence x≤2x\le 2.
04 / The boundary is included, so exactly the first two listed values work.
Worked example 2 · Test solutions and endpoint restrictions

A student includes x=2x=2 in the solution of x−2x−2≥0\dfrac{x-2}{x-2}\ge0. Which assessment is correct?

At the proposed value, both numerator and denominator are zero.
Division by zero is undefined; cancellation does not restore a missing input.
For all other real inputs the quotient equals 11, so all other inputs satisfy the inequality.

Read the graphs

Translate distance and tolerance descriptions

Read the shaded set, write one absolute-value inequality, and solve your inequality back into interval notation. Explain the endpoint symbols.
Solution on a number lineThe interval between endpoints -5 and 1 is shaded. Both endpoints are open and excluded.-7-5013
The interval between endpoints -5 and 1 is shaded. Both endpoints are open and excluded.

The midpoint of the two endpoints is −2-2; their half-separation is 33.
Shading between the endpoints means distance from the midpoint is less than 33.
Therefore ∣x+2∣<3|{x + 2}|\lt 3. The open endpoints match the strict comparison.
The equivalent set is (−5,1)(-5, 1).

Answer: ∣x+2∣<3|{x + 2}|\lt 3

15 questions covering every idea, with a worked answer for each.Start practice ↗
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.

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