Absolute Value Inequalities - Varsity Tutors

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Absolute Value Inequalities

Study Guide

Key Definition

An absolute value inequality is an inequality that includes an absolute value expression. For example, $|x| < a$ or $|x| > a$.

Important Notes

  • Absolute value inequalities can be solved by splitting them into two separate cases, one for the positive value and one for the negative.
  • These inequalities are often visualized on a number line, where the absolute value represents distance from zero.

Mathematical Notation

$|x|$ denotes the absolute value of $x$.In the context of inequalities, absolute value represents the distance of a number from zero.

Why It Works

The absolute value of a number is always positive or zero, hence when solving inequalities, it's necessary to consider both the positive and negative values.

Remember

When solving absolute value inequalities, always remember to consider both the positive and negative cases.

Quick Reference

Absolute Value Inequality:$|x| < a$ or $|x| > a$

Understanding Absolute Value Inequalities

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Video explanation of this concept

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Beginner Explanation

Absolute value of a number is its distance from zero. When we talk about absolute value inequalities, we are comparing this distance with another number.Now showing Beginner level explanation.

Practice Problems

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1

Quick Quiz

Single Choice QuizBeginner

Solve the absolute value inequality $|x - 3| < 7$.

A$-4 < x < 10$B$-10 < x < 4$C$3 < x < 10$D$-4 < x < 3$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Temperature Variation

The temperature in a city varies such that the difference in temperature from 25°C is less than or equal to 5°C. Express this situation as an absolute value inequality.Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

Solve the absolute value inequality $|2x - 5| > 7$.

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

Solve the absolute value inequality $|3x - 2| + 1 \geq 5$.

A$x \leq -\tfrac{2}{3}$ or $x \geq 2$B$x \geq 1$ or $x \leq 2$C$x \leq -1$ or $x \geq 2$D$x \geq -1$ or $x \leq 2$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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