Absolute value inequalities are inequalities that have an absolute value sign and a variable inside the absolute value. For example, |x| < 6 and |x - 1| > 2 are absolute value inequalities.
Let c represent a positive real number.
|x| ≥ c is equivalent to x ≥ c or x ≤ - c
|x| ≤ c is equivalent to -c ≤ x ≤ c
1.
Solve |x - 1| > 2
|x - 1| > 2 is equivalent to x - 1 > 2 or x - 1 < -2
Solve x - 1 > 2
x - 1 > 2
x - 1 + 1 > 2 + 1
x > 3
Solve x - 1 < -2
x - 1 + 1 < -2 + 1
x < - 1
Choose any number bigger than 3 or any number smaller than -1
2.
Solve |2x + 1| < 7
|2x + 1| < 7 is equivalent to -7 < 2x + 1 < 7
-7 - 1 < 2x + 1 - 1 < 7 - 1
-8 < 2x < 6
-8 / 2 < 2x / 2 < 6 / 2
-4 < x < 3
Choose any number between -4 and 3.
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