Q: Why does |x| < 4 become one "between" inequality?

Absolute value is distance from zero. Saying the distance is less than 4 means x is between -4 and 4, which is -4 < x < 4. Less than means between, greater than means outside.

Q: What if the absolute value equals a negative number?

A distance can never be negative, so |anything| = -5 or |anything| -5 is true for every real number.

Question 1

How do I solve an absolute value equation?

Accepted Answer

Isolate the absolute value, then split into two cases: the inside equals the positive number, or the inside equals the negative of it. |2x - 3| = 7 becomes 2x - 3 = 7 or 2x - 3 = -7, giving x = 5 or x = -2.

Question 2

Why does |x| < 4 become one "between" inequality?

Accepted Answer

Absolute value is distance from zero. Saying the distance is less than 4 means x is between -4 and 4, which is -4 < x < 4. Less than means between, greater than means outside.

Question 3

What if the absolute value equals a negative number?

Accepted Answer

Question 4

Do I flip the sign when the coefficient of x is negative?

Accepted Answer

Yes, in each case where you divide by the negative coefficient. |-2x + 1| ≥ 5 splits into -2x + 1 ≥ 5 or -2x + 1 ≤ -5, and dividing by -2 flips both.

Question 5

How do I check my answers?

Accepted Answer

Plug each solution back into the original absolute value expression. The calculator does this for equations and shows that both values give the same distance.