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 always a non-negative value.

Front

How do you solve an absolute value inequality of the form |x| ≤ a?

Back

To solve |x| ≤ a, you split it into two inequalities: -a ≤ x ≤ a.

Front

How do you solve an absolute value inequality of the form |x| > a?

Back

To solve |x| > a, you split it into two inequalities: x < -a or x > a.

Front

What does the inequality |x + 3| ≤ 2 represent on a number line?

Back

It represents all numbers x that are within 2 units of -3, which is the interval [-5, -1].

Front

What does the inequality |x + 3| > 8 represent on a number line?

Back

It represents all numbers x that are more than 8 units away from -3, which is the union of the intervals (-∞, -11) and (5, ∞).

Front

What is the solution to the inequality |x + 1| ≥ 3?

Back

The solution is x ≤ -4 or x ≥ 2.

Front

What does the inequality |n + 2| - 3 ≥ -6 simplify to?

Back

It simplifies to |n + 2| ≥ -3, which is true for all real numbers.

Front

How do you interpret the solution d ≤ -3 or d ≥ 1?

Back

It means that d can be any number less than or equal to -3, or any number greater than or equal to 1.

Front

What is the graphical representation of the solution to |x + 3| ≤ 2?

Back

The graphical representation is a closed interval on the number line from -5 to -1.

Front

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

Back

|x| < a means x is strictly between -a and a, while |x| ≤ a includes the endpoints -a and a.