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# Compound Inequalities

## Multiple inequalities associated by 'and' and 'or' statements

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Practice Compound Inequalities
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Compound Inequalities

What if you had an inequality with more than one inequality symbol, like $-4 < x < 5$ or $x > -2$ or $x < -7$ ? How could you graph such inequalities? After completing this Concept, you'll be able to graph compound inequalities like these on a number line.

### Guidance

In this section, we’ll solve compound inequalities—inequalities with more than one constraint on the possible values the solution can have.

There are two types of compound inequalities:

1. Inequalities joined by the word “and,” where the solution is a set of values greater than a number and less than another number. We can write these inequalities in the form “ $x>a$ and $x ,” but usually we just write “ $a .” Possible values for $x$ are ones that will make both inequalities true.
2. Inequalities joined by the word “or,” where the solution is a set of values greater than a number or less than another number. We write these inequalities in the form “ $x>a$ or $x .” Possible values for $x$ are ones that will make at least one of the inequalities true.

You might wonder why the variable $x$ has to be greater than one number and/or less than the other number; why can’t it be greater than both numbers, or less than both numbers? To see why, let’s take an example.

Consider the compound inequality “ $x>5$ and $x>3$ .” Are there any numbers greater than 5 that are not greater than 3? No! Since 5 is greater than 3, everything greater than 5 is also greater than 3. If we say $x$ is greater than both 5 and 3, that doesn’t tell us any more than if we just said $x$ is greater than 5. So this compound inequality isn’t really compound; it’s equivalent to the simple inequality $x > 5$ . And that’s what would happen no matter which two numbers we used; saying that $x$ is greater than both numbers is just the same as saying that $x$ is greater than the bigger number, and saying that $x$ is less than both numbers is just the same as saying that $x$ is less than the smaller number.

Compound inequalities with “or” work much the same way. Every number that’s greater than 3 or greater than 5 is also just plain greater than 3, and every number that’s greater than 3 is certainly greater than 3 or greater than 5—so if we say “ $x>5$ or $x>3$ ,” that’s the same as saying just “ $x>3$ .” Saying that $x$ is greater than at least one of two numbers is just the same as saying that $x$ is greater than the smaller number, and saying that $x$ is less than at least one of two numbers is just the same as saying that $x$ is less than the greater number.

Write and Graph Compound Inequalities on a Number Line

#### Example A

Write the inequalities represented by the following number line graphs.

a)

b)

c)

Solution

a) The solution graph shows that the solution is any value between -40 and 60, including -40 but not 60.

Any value in the solution set satisfies both $x \ge -40$ and $x<60$ .

This is usually written as $-40 \le x < 60$ .

b) The solution graph shows that the solution is any value greater than 1 (not including 1) or any value less than -2 (not including -2). You can see that there can be no values that can satisfy both these conditions at the same time. We write: $x>1$ or $x < -2$ .

c) The solution graph shows that the solution is any value greater than 4 (including 4) or any value less than -1 (including - 1). We write: $x \ge 4$ or $x \le -1$ .

#### Example B

Graph the following compound inequalities on a number line.

a) $-4 \le x \le 6$

b) $x < 0$ or $x > 2$

c) $x \ge -8$ or $x \le -20$

Solution

a) The solution is all numbers between -4 and 6, including both -4 and 6.

b) The solution is all numbers less than 0 or greater than 2, not including 0 or 2.

c) The solution is all numbers greater than or equal to -8 or less than or equal to -20.

Solve Compound Inequalities Using a Graphing Calculator (TI-83/84 family)

Graphing calculators can show you the solution to an inequality in the form of a graph. This can be especially useful when dealing with compound inequalities.

#### Example C

Solve the following inequalities using a graphing calculator.

a) $5x+2(x-3) \ge 2$

b) $7x-2 < 10x+1 < 9x+5$

c) $3x+2 \le 10$ or $3x+2 \ge 15$

Solution

a) Press the [Y=] button and enter the inequality on the first line of the screen.

(To get the $\ge$ symbol, press [TEST] [2nd] [MATH] and choose option 4.)

Then press the [GRAPH] button.

Because the calculator uses the number 1 to mean “true” and 0 to mean “false,” you will see a step function with the $y-$ value jumping from 0 to 1.

The solution set is the values of $x$ for which the graph shows $y=1$ —in other words, the set of $x-$ values that make the inequality true.

Note: You may need to press the [WINDOW] key or the [ZOOM] key to adjust the window to see the full graph.

The solution is $x>\frac{8}{7}$ , which is why you can see the $y-$ value changing from 0 to 1 at about 1.14.

b) This is a compound inequality: $7x-2 < 10x +1$ and $10x+1 < 9x+5$ . You enter it like this:

(To find the [AND] symbol, press [TEST] , choose [LOGIC] on the top row and choose option 1.)

The resulting graph should look like this:

The solution are the values of $x$ for which $y=1$ ; in this case that would be $-1 .

c) This is another compound inequality.

(To enter the [OR] symbol, press [TEST] , choose [LOGIC] on the top row and choose option 2.)

The resulting graph should look like this:

The solution are the values of $x$ for which $y=1$ --in this case, $x \le 2.7$ or $x \ge 4.3$ .

Watch this video for help with the Examples above.

### Vocabulary

• Compound inequalities combine two or more inequalities with “and” or “or.”
• “And” combinations mean that only solutions for both inequalities will be solutions to the compound inequality.
• “Or” combinations mean solutions to either inequality will also be solutions to the compound inequality.

### Guided Practice

1. Write the inequality represented by the following number line graph.

2. Graph the following compound inequality on a number line.

$-15 < x \le 85$

Solution

1. The solution graph shows that the solution is any value that is both less than 25 (not including 25) and greater than -25 (not including -25). Any value in the solution set satisfies both $x >-25$ and $x < 25.$

This is usually written as $-25 < x < 25.$

2. The solution is all numbers between -15 and 85, not including -15 but including 85.

### Explore More

Write the compound inequalities represented by the following graphs.

Graph the following compound inequalities on a number line.

1. $-2 \le x \le 20$
2. $x < 7$ or $x > 25$
3. $x \ge -100$ or $x \le -50$
4. $-1 < x < 200$
5. $2000 < x \le 2001$
6. $x \le 1.56$ or $x > 1.78$
7. $x > 0.0005$ or $x \le -0.03$