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

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

Back

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

Front

What does it mean if an absolute value inequality is written as |x| < a?

Back

It means that x is within a distance of a from zero, so -a < x < a.

Front

What does it mean if an absolute value inequality is written as |x| > a?

Back

It means that x is more than a distance of a from zero, so x < -a or x > a.

Front

Solve the inequality: |x - 3| < 5.

Back

The solution is -2 < x < 8.

Front

Solve the inequality: |2x + 1| ≥ 3.

Back

The solution is x ≤ -2 or x ≥ 1.

Front

What is the first step in solving an absolute value inequality?

Back

Isolate the absolute value expression on one side of the inequality.

Front

What is the difference between 'no solution' and 'all real numbers' in the context of inequalities?

Back

'No solution' means there are no values that satisfy the inequality, while 'all real numbers' means every real number satisfies the inequality.

Front

How do you graph the solution of an absolute value inequality?

Back

You graph the critical points found from solving the inequality and shade the appropriate regions based on the inequality sign.

Front

What is the solution to the equation |x + 4| = 2?

Back

The solutions are x = -2 and x = -6.