graphing 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, typically written in the form |expression| < c, |expression| > c, |expression| ≤ c, or |expression| ≥ c, where c is a constant.

Front

How do you graph an absolute value inequality?

Back

To graph an absolute value inequality, first graph the corresponding equation as a solid or dotted line depending on whether the inequality is inclusive (≤ or ≥) or exclusive (< or >). Then, shade the appropriate region above or below the line based on the inequality.

Front

What does it mean to shade above the line in a graph?

Back

Shading above the line in a graph indicates that the solutions to the inequality are greater than the values on the line. For example, for f(x) ≥ expression, you would shade above the line.

Front

What does it mean to shade below the line in a graph?

Back

Shading below the line in a graph indicates that the solutions to the inequality are less than the values on the line. For example, for f(x) ≤ expression, you would shade below the line.

Front

What is the difference between a solid line and a dotted line in graphing inequalities?

Back

A solid line indicates that the points on the line are included in the solution set (for ≤ or ≥ inequalities), while a dotted line indicates that the points on the line are not included (for < or > inequalities).

Front

How do you determine whether to use a solid or dotted line for an absolute value inequality?

Back

You use a solid line for inequalities that include equal to (≤ or ≥) and a dotted line for inequalities that do not include equal to (< or >).

Front

What is the general form of an absolute value inequality?

Back

The general form of an absolute value inequality is |ax + b| < c or |ax + b| > c, where a, b, and c are constants.

Front

How do you solve an absolute value inequality?

Back

To solve an absolute value inequality, you split it into two separate inequalities: one for the positive case and one for the negative case, then solve each inequality separately.

Front

What does the expression |x - 2| + 1 represent in an absolute value inequality?

Back

The expression |x - 2| + 1 represents the distance of x from 2, shifted up by 1 unit. It is often used in inequalities to define a range of values.

Front

What is the significance of the constant in an absolute value inequality?

Back

The constant in an absolute value inequality determines the threshold or boundary that the expression must satisfy, affecting the range of solutions.