Solving Absolute Value Equations

Front

What is an 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 an absolute value equation?

Back

To solve an absolute value equation, isolate the absolute value expression and then set up two separate equations: one for the positive case and one for the negative case.

Front

What does the equation |x| = a imply?

Back

The equation |x| = a implies that x can be either a or -a, provided that a is non-negative.

Front

What is the solution to |x - 3| = 0?

Back

The solution is x = 3, since the absolute value is zero only when the expression inside is zero.

Front

What does it mean if |x| = -a?

Back

If |x| = -a, where a is a positive number, there is no solution because absolute values cannot be negative.

Front

Solve for x: |x + 4| = 10.

Back

x + 4 = 10 or x + 4 = -10; thus, x = 6 or x = -14.

Front

What is the solution to |2x - 3| + 5 = 10?

Back

First, isolate the absolute value: |2x - 3| = 5. Then, solve 2x - 3 = 5 and 2x - 3 = -5, giving x = 4 and x = -1.

Front

What is the significance of the absolute value in real-world contexts?

Back

Absolute value is used to represent quantities that cannot be negative, such as distance, temperature differences, and financial losses.

Front

Solve for a: |2a - 10| = 6.

Back

2a - 10 = 6 or 2a - 10 = -6; thus, a = 8 or a = 2.

Front

What does the equation |−2n| + 10 = −50 indicate?

Back

This equation has no solution because the left side, being an absolute value plus a positive number, cannot equal a negative number.