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, expressing a range of values that a variable can take based on its distance from a certain point.

Front

How do you express 'within a certain distance' using absolute value inequalities?

Back

If a variable x is within a distance d from a point a, it can be expressed as |x - a| < d.

Front

What does the inequality |x - a| ≤ d represent?

Back

It represents that x is within d units of a, including the endpoints, meaning a - d ≤ x ≤ a + d.

Front

How do you solve the inequality |x - 6| ≤ 0.1?

Back

To solve |x - 6| ≤ 0.1, you set up the compound inequality: 6 - 0.1 ≤ x ≤ 6 + 0.1, which simplifies to 5.9 ≤ x ≤ 6.1.

Front

What is the solution to the inequality |x - 6| ≥ 0.1?

Back

The solution is x ≤ 5.9 or x ≥ 6.1, meaning x is either less than or equal to 5.9 or greater than or equal to 6.1.

Front

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

Back

The solution indicates the range of values that satisfy the inequality, showing how far a variable can deviate from a central value.

Front

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

Back

|x - a| < d means x is strictly within d units of a, while |x - a| ≤ d includes the endpoints, meaning x can be exactly d units away from a.

Front

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

Back

You graph the solution by marking the critical points on a number line and shading the appropriate region based on the inequality.

Front

What does the inequality |p + 6| ≥ 13 imply about p?

Back

It implies that p is either less than or equal to -19 or greater than or equal to 7, as it represents values that are at least 13 units away from -6.

Front

How do you solve the compound inequality -1 < r < 9?

Back

This compound inequality indicates that r can take any value between -1 and 9, not including -1 and 9.