Absolute Value Inequalities/Interval Notation

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 inequality |x| < a represent?

Back

It represents the set of all numbers x that are within a distance a from 0 on the number line.

Front

What does the inequality |x| > a represent?

Back

It represents the set of all numbers x that are more than a distance a from 0 on the number line.

Front

How do you solve the inequality |x + 4| + 8 > 2?

Back

Subtract 8 from both sides to get |x + 4| > -6. Since absolute values are always non-negative, this inequality is true for all real numbers.

Front

What is interval notation?

Back

Interval notation is a way of writing subsets of the real number line using intervals. For example, (a, b) represents all numbers between a and b, not including a and b.

Front

How do you express the solution x < -3 and x > -5/3 in interval notation?

Back

The solution can be expressed as (-∞, -3) ∪ (-5/3, ∞).

Front

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

Back

Dividing by -5 reverses the inequality: |x - 4| < -4. Since absolute values cannot be negative, there is no solution.

Front

How do you solve the inequality -5|-n + 7| ≥ -45?

Back

Divide by -5 (reversing the inequality): |-n + 7| ≤ 9. This leads to two inequalities: -9 ≤ -n + 7 ≤ 9, which simplifies to -2 ≤ n ≤ 16.

Front

What is the solution to the inequality |x + 5| < 9?

Back

This can be rewritten as -9 < x + 5 < 9, leading to -14 < x < 4.

Front

How do you solve |10 + 4x| < 14?

Back

This can be rewritten as -14 < 10 + 4x < 14, leading to -6 < x < 1.