Absolute Value Equations & Inequalities

Front

What is the definition of absolute value?

Back

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is denoted as |x|.

Front

How do you solve the absolute value equation |x| = a?

Back

To solve |x| = a, you set up two equations: x = a and x = -a.

Front

What does the inequality |x - a| < b represent?

Back

It represents the values of x that are within a distance b from a on the number line.

Front

What does the inequality |x - a| > b represent?

Back

It represents the values of x that are more than a distance b from a on the number line.

Front

How do you graph the inequality |x - 3| < 2?

Back

You find the interval by solving -2 < x - 3 < 2, which gives 1 < x < 5. Graph this as an open interval between 1 and 5.

Front

What is the difference between an 'and' and an 'or' problem in absolute value inequalities?

Back

An 'and' problem means both conditions must be true (overlapping intervals), while an 'or' problem means at least one condition must be true (separate intervals).

Front

Solve the inequality: |x + 4| ≤ 3.

Back

The solution is -7 ≤ x ≤ -1.

Front

What is the solution to the equation |2x - 1| = 9?

Back

The solutions are x = 5 and x = -4.

Front

When do you use an open dot when graphing inequalities?

Back

You use an open dot for < and > symbols, indicating that the endpoint is not included.

Front

When do you use a closed dot when graphing inequalities?

Back

You use a closed dot for ≤ and ≥ symbols, indicating that the endpoint is included.