Algebra GRE Strategy Guide by Manhattan Prep

Author:Manhattan Prep
Language: eng
Format: epub
Publisher: Manhattan Prep Publishing

So the two solutions to the original equation are x ≥ 4 and x ≤ −4. Here it is represented on a number line:

As before, any number that is covered by the black arrow will make the inequality true. Because of the absolute value, there are now two arrows instead of one, but nothing else has changed. Any number to the left of −4 will make the inequality true, as will any number to the right of 4.

Looking back at the inequality |x| ≥ 4, you can now interpret it in terms of distance. |x| ≥ 4 means “x is at least 4 units away from zero, in either direction.” The black arrows indicate all numbers for which that statement is true.

Example 2: |x + 3| < 5

Once again, the absolute value is already isolated on one side, so now you need to set up the two inequalities. The first inequality replaces the absolute value with the positive of what's inside, and the second replaces the absolute value with the negative of what's inside:

+ (x + 3) < 5 and − (x + 3) < 5

Next, isolate the variable in each equation:

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