Absolute Value Inequalities

Learn and apply absolute value inequalities using equations, examples, and interpretation.

Lesson 4 of 53 parts Practice Flashcards

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Inside Distance

Objective: Explain and use inside distance in an Algebra 1 problem.

For d>0, |x−c|<d represents values between c−d and c+d; strict endpoints are excluded. It excludes points exactly d away.

Worked examples

Try itExample 1.1: Which x satisfies |x−5|<4?Show answer

5. The center has distance zero, less than the positive bound. The answer is 5.

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Inside Distance

Objective: Explain and use inside distance in an Algebra 1 problem.

For d>0, |x−c|<d represents values between c−d and c+d; strict endpoints are excluded. It excludes points exactly d away.

Worked examples

Try itExample 1.1: Which x satisfies |x−5|<4?Show answer

5. The center has distance zero, less than the positive bound. The answer is 5.

Outside Distance

Objective: Explain and use outside distance in an Algebra 1 problem.

For d>0, |x−c|>d represents two outward rays: x<c−d or x>c+d. Or, since either ray works.

Worked examples

Try itExample 2.1: Which x satisfies |x−8|<2?Show answer

8. The center has distance zero, less than the positive bound. The answer is 8.

Boundary Checks

Objective: Explain and use boundary checks in an Algebra 1 problem.

Substituting endpoints and one value from each region helps verify the direction and whether equality is included. Strict and inclusive signs treat them differently.

Worked examples

Try itExample 3.1: Which x satisfies |x−11|<3?Show answer

11. The center has distance zero, less than the positive bound. The answer is 11.

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