Key Concepts

### Writing Solutions to Absolute Value Inequalities

For any positive value of *a *and *x,* a single variable, or any algebraic expression:

Absolute Value Inequality |
Equivalent Inequality |
Interval Notation |

[latex]\left|{ x }\right|\le{ a}[/latex] | [latex]{ -a}\le{x}\le{ a}[/latex] | [latex]\left[-a, a\right][/latex] |

[latex]\left| x \right|\lt{a}[/latex] | [latex]{ -a}\lt{x}\lt{ a}[/latex] | [latex]\left(-a, a\right)[/latex] |

[latex]\left| x \right|\ge{ a}[/latex] | [latex]{x}\le\text{−a}[/latex] or [latex]{x}\ge{ a}[/latex] | [latex]\left(-\infty,-a\right]\cup\left[a,\infty\right)[/latex] |

[latex]\left| x \right|\gt\text{a}[/latex] | [latex]\displaystyle{x}\lt\text{−a}[/latex] or [latex]{x}\gt{ a}[/latex] | [latex]\left(-\infty,-a\right)\cup\left(a,\infty\right)[/latex] |

## Glossary

**Union** The solution of a compound inequality that consists of two inequalities joined with the word *or* is the union of the solutions of each inequality.

**Intersection** The solution of a compound inequality that consists of two inequalities joined with the word* and *is the intersection of the solutions of each inequality. In other words, both statements must be true at the same time. The solution to an *and* compound inequality are all the solutions that the two inequalities have in common

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