Способи розв'язування нерівностей з модулем

The Rules

There are two rules to follow when you've got an inequality with an absolute value.

1. If your simplified inequality looks like $|x| > y$, then set up and solve two inequalities: $x > y$ or $x < -y$, the word "or" links the two inequalities.

2. If your simplified inequality looks like $|x| < y$, then set up and solve one inequality: $- y < x < y$

(Note: inequalities that use $≥$ (i.e., greater than or equal to) will follow Rule 1 and inequalities that use $≤$ (i.e., less than or equal to) will follow Rule 2.)

To review steps required to solve compound inequalities, check out this lesson.

Sometimes, it may be easier to imagine these shortcuts when you plot points on a number line. With really big numbers, though, that may be hard to do.

Note

In order for these rules to work, arrange the inequality to have the variable on the left side and the constant on the right side of the inequality symbol.

Also, always be mindful of negative numbers and positive numbers - this is the place where most students mess up when solving problems with a linear inequality.

This easy setup works for real numbers and variables.

Make sure you keep track of what are the absolute value bars and what is the number $1$!

Alright, let's get started!

Let's look at examples of both cases.

$$|2x + 4 | \ge 4$$

This inequality will follow Rule 1.

This inequality uses a $≥$ sign. It falls under the first rule.

Now we need to set up and solve two inequalities:

$$\begin{align} 2x+3\ge4 \text{ or } 2x+3\le-4 \end{align}$$

Now that we've got the two inequalities set up, we need to solve each one for $x$.

$$\begin{align} 2x+3\ge4 \\ 2x\ge 1 \\ x \ge \frac{1}{2} \end{align}$$

$$\begin{align} 2x+3\le -4 \\ 2x\le -7 \\ x \le \frac{-7}{2} \end{align}$$

You can plot both values on a number line, too.

You can also give your answer in interval notation. Check out this lesson to refresh on how to do that!

You can take this concept a step further and solve word problems or tackle piecewise functions.

(Videos) Solving Absolute Value Inequalities

by PatrickJMT

Method 1: GREATER THAN

When solving absolute value inequalities, what steps you take are different depending on whether the absolute value expression is GREATER THAN ($\ge, >$) or LESS THAN ($\le, <$) the expression on the other side of the inequality symbol. Patrick JMT explains both methods:

Summary

Here are the steps to solving an absolute value inequality when the side with the absolute value is GREATER than everything on the other side of the equation:

1. Isolate the absolute value stuff on one side of the equation.

2. Rewrite the inequality into the two options shown in the video ($A > B$ and $A < -B$)

3. Solve both equations. There's your answer!

Method 2: LESS THAN

Here are the steps to solving an absolute value inequality when the side with the absolute value is LESS than everything on the other side of the equation:

1. Isolate the absolute value stuff on one side of the equation.

2. Rewrite the inequality into the compound inequality shown in the video ($-B < A < B$)

3. Solve the inequality. Make sure you do every operation on every part of the equation--the left, the middle, and the right. There's your answer!

Solving Absolute Value Inequalities

There are two simple rules when tackling absolute value inequalities.

Techniques for solving inequalities extend to those involving powers and radicals.

Let's do a practice problem to test your knowledge on which rule to use!