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

What does the inequality |x| < a represent?

Back

It represents all values of x that are within a distance a from 0, meaning -a < x < a.

Front

What does the inequality |x| > a represent?

Back

It represents all values of x that are more than a distance a from 0, meaning x < -a or x > a.

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

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

Back

Add 8 to both sides to get |x + 4| > 10. This leads to two cases: x + 4 > 10 or x + 4 < -10, resulting in x > 6 or x < -14.

Front

How do you solve the inequality |c + 4| < 5?

Back

This leads to -5 < c + 4 < 5. Subtracting 4 gives -9 < c < 1.

Front

How do you solve the inequality |p + 3| ≥ 10?

Back

This leads to two cases: p + 3 ≤ -10 or p + 3 ≥ 10, resulting in p ≤ -13 or p ≥ 7.

Front

How do you solve the inequality |y - 4| ≥ 8?

Back

This leads to two cases: y - 4 ≤ -8 or y - 4 ≥ 8, resulting in y ≤ -4 or y ≥ 12.

Front

What is the graphical representation of |x| < a?

Back

It is represented by an open interval on the number line from -a to a.

Front

What is the graphical representation of |x| > a?

Back

It is represented by two rays extending from -∞ to -a and from a to ∞ on the number line.