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# Compound Inequalities

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

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Compound Inequalities

What if you had an inequality with more than one inequality symbol, like 4<x<5\begin{align*}-4 < x < 5\end{align*} or x>2\begin{align*}x > -2\end{align*} or x<7\begin{align*}x < -7\end{align*}? 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\begin{align*}x>a\end{align*} and x<b\begin{align*}x,” but usually we just write “a<x<b\begin{align*}a.” Possible values for x\begin{align*}x\end{align*} 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\begin{align*}x>a\end{align*} or x<b\begin{align*}x.” Possible values for x\begin{align*}x\end{align*} are ones that will make at least one of the inequalities true.

You might wonder why the variable x\begin{align*}x\end{align*} 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\begin{align*}x>5\end{align*} and x>3\begin{align*}x>3\end{align*}.” 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\begin{align*}x\end{align*} is greater than both 5 and 3, that doesn’t tell us any more than if we just said x\begin{align*}x\end{align*} is greater than 5. So this compound inequality isn’t really compound; it’s equivalent to the simple inequality x>5\begin{align*}x > 5\end{align*}. And that’s what would happen no matter which two numbers we used; saying that x\begin{align*}x\end{align*} is greater than both numbers is just the same as saying that x\begin{align*}x\end{align*} is greater than the bigger number, and saying that x\begin{align*}x\end{align*} is less than both numbers is just the same as saying that x\begin{align*}x\end{align*} 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\begin{align*}x>5\end{align*} or x>3\begin{align*}x>3\end{align*},” that’s the same as saying just “x>3\begin{align*}x>3\end{align*}.” Saying that x\begin{align*}x\end{align*} is greater than at least one of two numbers is just the same as saying that x\begin{align*}x\end{align*} is greater than the smaller number, and saying that x\begin{align*}x\end{align*} is less than at least one of two numbers is just the same as saying that x\begin{align*}x\end{align*} 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 x40\begin{align*}x \ge -40\end{align*} and x<60\begin{align*}x<60\end{align*}.

This is usually written as 40x<60\begin{align*}-40 \le x < 60\end{align*}.

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\begin{align*}x>1\end{align*} or x<2\begin{align*}x < -2\end{align*}.

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: x4\begin{align*}x \ge 4\end{align*} or x1\begin{align*}x \le -1\end{align*}.

#### Example B

Graph the following compound inequalities on a number line.

a) 4x6\begin{align*}-4 \le x \le 6\end{align*}

b) x<0\begin{align*}x < 0\end{align*} or x>2\begin{align*}x > 2\end{align*}

c) x8\begin{align*}x \ge -8\end{align*} or x20\begin{align*}x \le -20\end{align*}

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(x3)2\begin{align*}5x+2(x-3) \ge 2\end{align*}

b) 7x2<10x+1<9x+5\begin{align*}7x-2 < 10x+1 < 9x+5\end{align*}

c) 3x+210\begin{align*}3x+2 \le 10\end{align*} or 3x+215\begin{align*}3x+2 \ge 15\end{align*}

Solution

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

(To get the \begin{align*}\ge\end{align*} 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\begin{align*}y-\end{align*}value jumping from 0 to 1.

The solution set is the values of x\begin{align*}x\end{align*} for which the graph shows y=1\begin{align*}y=1\end{align*}—in other words, the set of x\begin{align*}x-\end{align*}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>87\begin{align*}x>\frac{8}{7}\end{align*}, which is why you can see the y\begin{align*}y-\end{align*}value changing from 0 to 1 at about 1.14.

b) This is a compound inequality: \begin{align*}7x-2 < 10x +1\end{align*} and \begin{align*}10x+1 < 9x+5\end{align*}. 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 \begin{align*}x\end{align*} for which \begin{align*}y=1\end{align*}; in this case that would be \begin{align*}-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 \begin{align*}x\end{align*} for which \begin{align*}y=1\end{align*}--in this case, \begin{align*}x \le 2.7\end{align*} or \begin{align*}x \ge 4.3\end{align*}.

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.

\begin{align*}-15 < x \le 85\end{align*}

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 \begin{align*}x >-25\end{align*} and \begin{align*}x < 25.\end{align*}

This is usually written as \begin{align*}-25 < x < 25.\end{align*}

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

### Practice

Write the compound inequalities represented by the following graphs.

Graph the following compound inequalities on a number line.

1. \begin{align*}-2 \le x \le 20\end{align*}
2. \begin{align*}x < 7\end{align*} or \begin{align*}x > 25\end{align*}
3. \begin{align*}x \ge -100\end{align*} or \begin{align*}x \le -50\end{align*}
4. \begin{align*}-1 < x < 200\end{align*}
5. \begin{align*}2000 < x \le 2001\end{align*}
6. \begin{align*}x \le 1.56\end{align*} or \begin{align*}x > 1.78\end{align*}
7. \begin{align*}x > 0.0005\end{align*} or \begin{align*}x \le -0.03\end{align*}