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# Compound Inequalities

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

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Compound Inequalities

Do you know how to graph an inequality with more than one inequality symbol, like 4<x<5\begin{align*}-4 < x < 5\end{align*}

### Compound Inequalities

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

Write the inequalities represented by the following number line graphs.

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*}.

#### Graph the following compound inequalities on a number line.

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

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

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

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

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

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.

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

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

This is a compound inequality: 7x2<10x+1\begin{align*}7x-2 < 10x +1\end{align*} and 10x+1<9x+5\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 x\begin{align*}x\end{align*} for which y=1\begin{align*}y=1\end{align*}; in this case that would be 1<x<4\begin{align*}-1.

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

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\begin{align*}x\end{align*} for which y=1\begin{align*}y=1\end{align*}--in this case, x2.7\begin{align*}x \le 2.7\end{align*} or x4.3\begin{align*}x \ge 4.3\end{align*}.

### Examples

#### Example 1

Write the inequality represented by the following number line graph.

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

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

#### Example 2

Graph the following compound inequality on a number line.

15<x85\begin{align*}-15 < x \le 85\end{align*}

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

### Review

Write the compound inequalities represented by the following graphs.

Graph the following compound inequalities on a number line.

1. 2x20\begin{align*}-2 \le x \le 20\end{align*}
2. x<7\begin{align*}x < 7\end{align*} or x>25\begin{align*}x > 25\end{align*}
3. x100\begin{align*}x \ge -100\end{align*} or x50\begin{align*}x \le -50\end{align*}
4. 1<x<200\begin{align*}-1 < x < 200\end{align*}
5. 2000<x2001\begin{align*}2000 < x \le 2001\end{align*}
6. x1.56\begin{align*}x \le 1.56\end{align*} or x>1.78\begin{align*}x > 1.78\end{align*}
7. x>0.0005\begin{align*}x > 0.0005\end{align*} or x0.03\begin{align*}x \le -0.03\end{align*}

To see the Review answers, open this PDF file and look for section 6.5.

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