Algebra 1 Section 4.7

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Absolute Value The absolute value of a number refers to its distance from the origin. This will help us interpret absolute value inequalities.

Absolute Value Inequalities To find the solutions of |x| < 3, we must find the numbers whose distance from the origin is less than 3. Examples: -2, 1.5, 0, -0.5

Absolute Value Inequalities This is equivalent to: x > -3 and x < 3 -3 < x < 3

Absolute Value Inequalities The solution to the absolute value inequality |x| ≥ -3 includes all numbers whose distance from the origin is greater than or equal to 3.

Absolute Value Inequalities This is equivalent to: x ≤ -3 or x ≥ 3

Absolute Value Inequalities If |x| > c, then x < -c or x > c If |x| < c, then x > -c and x < c

Example 1 |x| > 1 x < -1 or x > 1

Example 1 |x| ≤ 4 -4 ≤ x ≤ 4

Example 2 |x + 5| ≤ 10 x + 5 ≥ -10 and x + 5 ≤ 10 x ≥ -15 and x ≤ 5

Example 3 |-3x + 6| > 18 -3x + 6 > 18 or -3x + 6 < -18

Absolute Value Inequalities Be sure the absolute value is isolated before you write the two separate inequalities.

Absolute Value Inequalities Type of Solution Set Operation Inequality conjunction (and) < or ≤ intersection disjunction (or) > or ≥ union

Example 4 |t – 98.6| ≤ 2 optimal temperature within 2°

Example 4 |t – 98.6| ≤ 2 t – 98.6 ≤ 2 and t – 98.6 ≥ -2

Homework: pp. 175-176