explain absolute value equation and inequalities with examples

Question

explain how to use absolute value equation and inequalities with examples in college algebra

Answer 1

Absolute value equations and inequalities can seem confusing at first, but they become much easier when you think of absolute value as distance from zero. 

For example, ∣x∣=5 means that x is 5 units away from zero, so x=5 or x=−5.

For an absolute value equation like ∣x−2∣=4, think of it as asking, “What numbers are 4 units away from 2?” The answers are x=6 and x=−2.

For inequalities, the same distance idea helps. If ∣x∣<3, x must be less than 3 units away from zero, so the answer is −3<x<3. If ∣x∣>3, x must be more than 3 units away from zero, so x<−3 or x>3.

A good way to remember it is: “less than” means you stay between the two numbers, while “greater than” means you move outside them.

Answer 2

Let's take the inequality |x + 2| > 5 . Since we're taking the absolute value of x+2, then x+2 could be positive or negative. Therefore, we need to consider both situations:

x + 2 > 5    or x + 2 < -5

WAIT! Why did the direction of the sign change???

We could also look at this as:

x + 2 > 5   or (-1)(x + 2) > 5  (the negative of x+2 is greater than 5) . In the second inequality, once we divide by (-1), the direction of the inequality changes. 

Now we solve each one separately:

x + 2 > 5    ===> x > 3

X + 2 < -5   ===> x < -7

so our solution set is x > 3 or x < -7

Why is it OR and not AND? Because our solutions go different directions on the number line, and both cannot be true at the same time.