2.6 Solving Absolute Value Inequalities

Front

What is an absolute value inequality?

Back

An absolute value inequality is an inequality that contains an absolute value expression, which measures the distance of a number from zero on the number line.

Front

How do you solve the inequality |x + 3| ≤ 2?

Back

To solve |x + 3| ≤ 2, split it into two inequalities: -2 ≤ x + 3 ≤ 2. This simplifies to -5 ≤ x ≤ -1.

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What does the solution x ≤ -4 or x ≥ 2 represent in the inequality |x + 1| ≥ 3?

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It represents the values of x that are either less than or equal to -4 or greater than or equal to 2, indicating that the distance from -1 is at least 3.

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What is the solution to the inequality |x| ≤ 3.5?

Back

The solution is -3.5 ≤ x ≤ 3.5, meaning x is within 3.5 units from zero.

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What does it mean if an absolute value inequality has no solution?

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It means that there are no values of the variable that satisfy the inequality, such as |3c - 5| < -2.

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How do you interpret the inequality |x - 4| < 5?

Back

It means that x is within 5 units of 4, leading to the solution -1 < x < 9.

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What is the first step in solving |2x + 1| ≥ 4?

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The first step is to split it into two cases: 2x + 1 ≥ 4 and 2x + 1 ≤ -4.

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What does the inequality |x + 2| ≤ 0 imply?

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It implies that x + 2 must equal 0, leading to the solution x = -2.

Front

How do you graph the solution of |x| < 3?

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You graph the interval (-3, 3) on a number line, indicating all numbers between -3 and 3, not including -3 and 3.

Front

What is the significance of the 'or' in the solution x ≤ -4 or x ≥ 2?

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The 'or' indicates that either condition can be true for the solution set, meaning x can be in either range.