Absolute Value Equations and 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

What does the equation |x - a| = b represent?

Back

It represents two possible equations: x - a = b and x - a = -b, which can be solved to find the values of x.

Front

How do you solve the equation |x - 5| = 3?

Back

Split into two equations: x - 5 = 3 and x - 5 = -3. Solutions are x = 8 and x = 2.

Front

What is the meaning of the inequality |x - a| ≤ b?

Back

It means that x is within b units of a, or a - b ≤ x ≤ a + b.

Front

What is the difference between |x| < a and |x| ≤ a?

Back

|x| < a means x is strictly less than a units from zero (open interval), while |x| ≤ a includes the endpoints (closed interval).

Front

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

Back

Graph the interval (1, 5) on a number line, using open circles at 1 and 5.

Front

What is the solution set for the inequality |x + 4| ≥ 3?

Back

The solution set is x ≤ -7 or x ≥ -1.

Front

What does it mean to use an open circle when graphing inequalities?

Back

An open circle indicates that the endpoint is not included in the solution set (for < or >).

Front

How do you interpret the absolute value equation |v - 20| = 0.03?

Back

It means the value v is within 0.03 ounces of 20, so v can be 19.97 or 20.03.

Front

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

Back

The solutions are x = -1 and x = 4.