Absolute Value 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 inequality |x + 3| > 8?

Back

Split into two cases: x + 3 > 8 or x + 3 < -8. This leads to x > 5 or x < -11.

Front

What is the solution to the inequality |x + 4| - 8 > 2?

Back

Rearranging gives |x + 4| > 10. This leads to x > 6 or x < -14.

Front

Back

It represents the values of x for which the quadratic expression is positive. The solution is x < -4 or x > -2.

Front

Back

Factor to get (x - 1)(x - 3) \ge 0. The solution is x \le 1 or x \ge 3.

Front

What is the process for solving absolute value inequalities?

Back

1. Isolate the absolute value expression. 2. Set up two separate inequalities based on the definition of absolute value. 3. Solve each inequality.

Front

What is the solution to the inequality -5|x - 4| > -20?

Back

Divide by -5 (reversing the inequality): |x - 4| < 4. This leads to 0 < x < 8.

Front

What does the expression |x| < a mean?

Back

It means that x is within a distance a from 0, or -a < x < a.

Front

What is the graphical representation of an absolute value inequality?

Back

It is represented on a number line with open or closed circles indicating the solutions, depending on whether the inequality is strict (<, >) or inclusive (≤, ≥).

Front

What is the difference between strict and non-strict inequalities?

Back

Strict inequalities (<, >) do not include the boundary points, while non-strict inequalities (≤, ≥) do include them.