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# 6.4: Compound Inequalities

Difficulty Level: At Grade Created by: CK-12

Inequalities that relate to the same topic can be written as a compound inequality. A compound inequality involves the connecting words “and” and “or.”

The word and in mathematics means the intersection between the sets.

“What the sets have in common.”

The word or in mathematics means the union of the sets.

“Combining both sets into one large set.”

## Inequalities Involving “And”

Consider, for example, the speed limit situation from the previous lesson. Using interval notation, the solutions to this situation can be written as [0, 65]. As an inequality, what is being said it this:

The speed must be at least 0 mph and at most 65 mph.

Using inequalities to represent “at least” and “at most,” the following sentences are written:

\begin{align*}s \ge 0 \ \text{and} \ s \le 65\end{align*}

This is an example of a compound inequality. It can be shortened by writing:

\begin{align*}0 \le s \le 65\end{align*}

Example: Graph the solutions to \begin{align*}-40 \le y < 60\end{align*}.

Solution: Color in a circle above –40 to represent “less than or equal to.” Draw an uncolored circle above 60. The variable is placed between these two values, so the solutions occur between these two numbers.

## Inequalities Involving “Or”

A restaurant offers discounts to children 3 years or younger or to adults over 65. Graph the possible ages eligible to receive the discount.

Begin by writing an inequality to represent each piece. “3 years or younger” means you must be born but must not have celebrated your fourth birthday.

\begin{align*}0 \le a < 4\end{align*}

“Adults over 65” implies \begin{align*}a > 65\end{align*}.

The word or between the phrases allows you to graph all the possibilities on one number line.

## Solving “And” Compound Inequalities

When we solve compound inequalities, we separate the inequalities and solve each of them separately. Then, we combine the solutions at the end.

To solve \begin{align*}3x-5 < x + 9 \le 5x + 13\end{align*}, begin by separating the inequalities.

\begin{align*}3x-5 & < x+9 &&&& \ x+9 \le 5x+13\\ 2x & < 14 && \text{and} && \quad -4 \le 4x\\ x & < 7 && && \quad -1 \le x \ \text{or } x\ge -1\end{align*}

The answers are \begin{align*}x<7\end{align*} and \begin{align*}x \ge -1\end{align*} and can be written as \begin{align*}-1 \le x < 7\end{align*}. You graph the solutions that satisfy both inequalities.

## Solving “Or” Compound Inequalities

To solve an “or” compound inequality, separate the individual inequalities. Solve each separately. Then combine the solutions to finish the problem.

To solve \begin{align*}9-2x \le 3\end{align*} or \begin{align*}3x+10 \le 6-x\end{align*}, begin by separating the inequalities.

\begin{align*}9-2x & \le 3 &&&& 3x+10 \le 6-x\\ -2x & \le -6 && \text{or} && \ \qquad 4x \le -4\\ x & \ge 3 &&&& \qquad \ \ x \le -1\end{align*}

The answers are \begin{align*}x \ge 3\end{align*} or \begin{align*}x \le -1\end{align*}.

## Using a Graphing Calculator to Solve Compound Inequalities

As you have seen in previous lessons, graphing calculators can be used to solve many complex algebraic sentences.

Example: Solve \begin{align*}7x-2 < 10x+1 < 9x+5\end{align*} using a graphing calculator.

Solution: This is a compound inequality \begin{align*}7x-2 < 10x+1\end{align*} and \begin{align*}10x+1 < 9x+5.\end{align*}

To enter a compound inequality:

Press the [Y=] button.

The inequality symbols are found by pressing [TEST] [2nd] [MATH]

Enter the inequality as:

\begin{align*}Y_1 = (7x-2 < 10x+1) \ AND \ (10x+1 < 9x+5)\end{align*}

To enter the [AND] symbol, press [TEST]. Choose [LOGIC] on the top row and then select option 1.

The resulting graph looks as shown below.

The solutions are the values of \begin{align*}x\end{align*} for which \begin{align*}y=1\end{align*}.

In this case, \begin{align*}-1 < x < 4\end{align*} .

## Solve Real-World Compound Inequalities

Example: The speed of a golf ball in the air is given by the formula \begin{align*}v=-32t+80\end{align*}, where \begin{align*}t\end{align*} is the time since the ball was hit. When is the ball traveling between 20 ft/sec and 30 ft/sec inclusive?

Solution: We want to find the times when the ball is traveling between 20 ft/sec and 30 ft/sec inclusive. Begin by writing the inequality to represent the unknown values, \begin{align*}20 \le v \le 30\end{align*}.

Replace the velocity formula \begin{align*}v=-32t+80\end{align*}, with the minimum and maximum values.

\begin{align*}20 \le -32t + 80 \le 30\end{align*}

Separate the compound inequality and solve each separate inequality.

\begin{align*}20 & \le -32t + 80 &&&& -32t + 80 \le 30\\ 32t & \le 60 && \text{and} && \qquad \qquad 50 \le 32t\\ t & \le 1.875 &&&& \qquad \quad \ 1.56 \le t\end{align*}

\begin{align*}1.56 \le t \le 1.875\end{align*}. Between 1.56 and 1.875 seconds, the ball is traveling between 20 ft/sec and 30 ft/sec.

Inequalities can also be combined with dimensional analysis.

Example: William’s pick-up truck gets between 18 and 22 miles per gallon of gasoline. His gas tank can hold 15 gallons of gasoline. If he drives at an average speed of 40 miles per hour, how much driving time does he get on a full tank of gas?

Solution: Use dimensional analysis to get from time per tank to miles per gallon.

\begin{align*}\frac{t \ \cancel{hour}}{1 \ \cancel{tank}} \times \frac{1 \ \cancel{tank}}{15 \ gallons} \times \frac{40 \ miles}{1 \ \cancel{hour}} = \frac{40t}{45} \frac{miles}{gallon}\end{align*}

Since the truck gets between 18 to 22 miles/gallon, you can write a compound inequality.

\begin{align*}18 \le \frac{40t}{15} \le 22\end{align*}

Separate the compound inequality and solve each inequality separately.

\begin{align*}18 & \le \frac{40t}{15} &&&& \frac{40t}{15} \le 22\\ 270 & \le 40t && \text{and} && \ 40t \le 330\\ 6.75 & \le t &&&& \quad \ t \le 8.25\end{align*}

William can drive between 6.75 and 8.25 hours on a full tank of gas.

## Practice Set

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set.  However, the practice exercise is the same in both.

1. Describe the solution set to a compound inequality joined by the word “and.”
2. How would your answer to question #1 change if the joining word was “or.”
3. Write the process used to solve a compound inequality.

Write the compound inequalities represented by the following graphs.

Graph each compound inequality on a number line.

1. \begin{align*}-4 \le x \le 6\end{align*}
2. \begin{align*}x < 0\end{align*} or \begin{align*}x > 2\end{align*}
3. \begin{align*}x \ge -8\end{align*} or \begin{align*}x \le -20\end{align*}
4. \begin{align*}-15 < x \le 85\end{align*}

In 15 – 30, solve the following compound inequalities and graph the solution on a number line.

1. \begin{align*}-5 \le x-4 \le 13\end{align*}
2. \begin{align*}-2 < 4x -5 \le 11\end{align*}
3. \begin{align*}\frac{x-2}{6} \le 2x-4\end{align*} or \begin{align*}\frac{x-2}{6} > x+5\end{align*}
4. \begin{align*}1 \le 3x+4 \le 4\end{align*}
5. \begin{align*}-12 \le 2 - 5x \le 7\end{align*}
6. \begin{align*}\frac{3}{4} \le 2x+9 \le \frac{3}{2}\end{align*}
7. \begin{align*}-2\frac{2x-1}{3} < -1\end{align*}
8. \begin{align*}5x+2(x-3)\ge 2\end{align*}
9. \begin{align*}3x+2 \le 10\end{align*} or \begin{align*}3x+2 \ge 15\end{align*}
10. \begin{align*}4x-1 \ge 7\end{align*} or \begin{align*}\frac{9x}{2} < 3\end{align*}
11. \begin{align*}3-x < -4\end{align*} or \begin{align*}3-x > 10\end{align*}
12. \begin{align*}\frac{2x+3}{4} < 2\end{align*} or \begin{align*}-\frac{x}{5}+3\frac{2}{5}\end{align*}
13. \begin{align*}2x - 7 \le -3\end{align*} or \begin{align*}2x - 3 > 11\end{align*}
14. \begin{align*}-6d>48\end{align*} or \begin{align*}10+d>11\end{align*}
15. \begin{align*}6+b<8\end{align*} or \begin{align*}b+6 \ge 6\end{align*}
16. \begin{align*}4x+3 \le 9\end{align*} or \begin{align*}-5x+4 \le -12\end{align*}
17. Using the golf ball example, find the times in which the velocity of the ball is between 50 ft/sec and 60 ft/sec.
18. Using the pick-up truck example, suppose William’s truck has a dirty air filter, causing the fuel economy to be between 16 and 18 miles per gallon. How many hours can William drive on a full tank of gas using this information?
19. To get a grade of B in her Algebra class, Stacey must have an average grade greater than or equal to 80 and less than 90. She received the grades of 92, 78, and 85 on her first three tests. Between which scores must her grade fall on her last test if she is to receive a grade of B for the class?

Mixed Review

1. Solve the inequality and write its solution in interval notation: \begin{align*}\frac{x+3}{2}<-4\end{align*}.
2. Graph \begin{align*}2x-2y=6\end{align*} using its intercepts.
3. Identify the slope and \begin{align*}y-\end{align*}intercept of \begin{align*}y+1=\frac{2}{5}(x-5)\end{align*}.
4. A yardstick casts a one-foot shadow. What is the length of the shadow of a 16-foot tree?
5. George rents videos through a mail-order company. He can get 16 movies each month for $16.99. Sheri rents videos through instant watch. She pays$1.99 per movie. When will George pay less than Sheri?
6. Evaluate: \begin{align*}-2\frac{1}{5} \div 1\frac{3}{4}\end{align*}.
7. Find a line parallel to \begin{align*}y=5x-2\end{align*} containing (1, 1).

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