ABSOLUTE VALUE INEQUALITIES

Front

What is an absolute value inequality?

Back

An absolute value inequality is an inequality that involves the absolute value of a variable expression, indicating the distance of the expression from zero on a number line.

Front

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

Back

To solve |x + 4| < 5, split it into two inequalities: -5 < x + 4 < 5. This simplifies to -9 < x < 1.

Front

What is the solution to the inequality |p + 3| ≥ 10?

Back

The solution is p ≤ -13 or p ≥ 7.

Front

How do you interpret the inequality |x - 6| ≤ 0.1 in a real-world context?

Back

This inequality indicates that the value of x (the radius of the gears) must be within 0.1 inches of 6 inches, meaning it can range from 5.9 to 6.1 inches.

Front

What does the inequality |x + 3| ≤ 2 represent?

Back

It represents the range of x values that are within 2 units of -3, which is -5 ≤ x ≤ -1.

Front

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

Back

The first step is to isolate the absolute value expression on one side of the inequality.

Front

What does the solution |x - 4| < 3 look like on a number line?

Back

The solution represents all x values between 1 and 7, which can be shown as an open interval (1, 7) on a number line.

Front

How do you express the solution to |x| < a, where a > 0?

Back

The solution is -a < x < a.

Front

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

Back

|x| < a means x is within a distance a from 0, while |x| ≥ a means x is at least a distance a from 0.

Front

How do you solve the inequality 3|d - 4| ≥ 12?

Back

First, divide by 3 to get |d - 4| ≥ 4, then split into two inequalities: d - 4 ≥ 4 or d - 4 ≤ -4, leading to d ≥ 8 or d ≤ 0.