Solving Absolute-Value Inequalities

Front

What is an absolute value inequality?

Back

An absolute value inequality is an inequality that contains an absolute value expression, which measures the distance of a number from zero on the number line.

Front

How do you solve the inequality |x + 1| ≥ 3?

Back

To solve |x + 1| ≥ 3, split it into two cases: x + 1 ≥ 3 or x + 1 ≤ -3. This gives x ≥ 2 or x ≤ -4, so the solution is x ≤ -4 or x ≥ 2.

Front

What does the solution x ≤ -4 or x ≥ 2 represent on a number line?

Back

It represents all numbers less than or equal to -4 and all numbers greater than or equal to 2.

Front

How do you solve the inequality |2w - 1| < 11?

Back

To solve |2w - 1| < 11, split it into two cases: 2w - 1 < 11 and 2w - 1 > -11. This gives -5 < w < 6.

Front

What is the solution set for the inequality |x - 2| > 3?

Back

The solution set is x < -1 or x > 5.

Front

What does the inequality |x - 2| > 3 indicate about the values of x?

Back

It indicates that x is either more than 3 units away from 2 on the number line, meaning it can be less than -1 or greater than 5.

Front

How do you interpret the inequality -5|x - 4| > -20?

Back

First, divide both sides by -5 (remember to flip the inequality), resulting in |x - 4| < 4, which means 0 < x < 8.

Front

What is the significance of the absolute value in inequalities?

Back

The absolute value indicates the distance from zero, allowing us to express conditions where a variable can be either above or below a certain threshold.

Front

What is the general approach to solving absolute value inequalities?

Back

1. Isolate the absolute value expression. 2. Split into two cases (for < or >). 3. Solve each case. 4. Combine the solutions.

Front

What does the solution x < 8 and x > 0 mean in the context of an absolute value inequality?

Back

It means that x can take any value between 0 and 8, not including 0 and 8.