Analyzing One-Variable Inequalities in Context

Similar to equations, inequalities can be used to represent real-world relationships. The method used to solve a problem with an inequality is the same as when using an equation. Namely, an inequality representing the relationship between quantities can be written and solved.

Method

Interpreting Inequalities

When interpreting a one-variable inequality, it is necessary to connect the terms on both sides of the inequality to specific quantities. It can be helpful to begin with the simplest side. Try to connect that to information in the text. When both sides of the inequality are expressed in words, the solution of the inequality can be interpreted.

Interpret and solve the inequality

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Exercise

George wants to run a $10$ mile race in under $90$ minutes. Halfway through the race, at mile $5,$ George’s personal trainer, Haile, gives him a note that reads $41 + 5 x < 90 .$ What is Haile trying to tell George? Interpret and solve the inequality.

Show Solution

Solution

Example

Interpretation

To begin, we can interpret each side of the inequality separately. Let's begin with the right-hand side since it only has one term. We know that George wants to finish the race in less than $90$ minutes. Thus, the right-hand side of the inequality represents George's maximum time. The left-hand side, $41 + 5 x$ is more complicated. We can think of George's total time in terms of the time it takes to run the first half and the second half separately. When Haile passes George the note, the first half of the race is completed. Thus, $41$ represents the number of minutes it took George to run the first half of the race. The term $5 x$ represents the time George can take to run the second half of the race while still meeting his goal. Since $5$ is the number of miles left to run, $x$ must be the maximum time per mile George can take.

Example

Solving the inequality

Solving the inequality will allow us to determine the maximum time per mile George can run during the second half of the race to meet his goal. To solve, we must isolate

x .

41 + 5 x < 90

SubIneq

LHS - 41 < RHS - 41

41 + 5 x - 41 < 90 - 41

SubTermsSubtract terms

5 x < 49

DivIneq

LHS / 5 < RHS / 5

\frac{5 x}{5} < \frac{4 9}{5}

SimpQuotSimplify quotient

x < \frac{4 9}{5}

UseCalcUse a calculator

x < 9.8

The solution to the inequality is

x < 9.8 .

Therefore, if George wants to meet his goal, his maximum time per mile in the second half of the race must be less than

9.8

minutes.

Method

Modeling with Inequalities

Inequalities can be used as mathematical models when real-life relationship need to be analyzed. What follows is one method of using mathematical models to solve problems.

Bear cubs are born during winter and first come out of their den in spring. Suppose spring begins once the mean temperature during a five day period is higher than $4 1^{\circ} F .$ Following a period of cold weather, four warmer days were registered in the forest. The highest temperatures these days were $3 7^{\circ} F,$ $4 2^{\circ} F,$ $3 8^{\circ} F,$ and $4 4^{\circ} F .$ How warm must the fifth day be for the cubs to come out of their den?

1

Make sense of the given information

Here, highlight the important information from the situation.

The temperatures the first four days were $37^{\circ} F,$ $42^{\circ} F,$ $38^{\circ} F,$ and $44^{\circ} F .$
The mean temperature for five consecutive days must be higher than $41^{\circ} F .$

2

Define variable

Typically, a variable is used to represent an unknown quantity in a situation. Here, the unknown quantity is the highest temperature on the fifth day. The variable $T$ can be used to represent the highest temperature that day.

3

Relate quantities

Next, it is necessary to understand how the different quantities in the problem relate. The mean temperature over five days is the sum of the highest temperatures each day, divided by the number of days, $5 .$ Since the mean must be greater than $4 1^{\circ} F,$ the following inequality can be written. $mean > 41 .$

4

Create inequality

Creating the inequality involves translating the relationship from Step $3$ into symbols. The mean of a data set can be calculated using $mean = \frac{sum of data points}{number of data points} .$ Thus, the mean temperature over five days can be found by adding the given temperatures and dividing by the $5,$ the number of temperatures. $mean = \frac{3 7 + 4 2 + 3 8 + 4 4 + T}{5}$ Replace the right-hand side of this expression into the inequaltiy from Step 3. $\frac{3 7 + 4 2 + 3 8 + 4 4 + T}{5} > 41$

5

Solve the inequality

Solve the created inequality to determine the unknown value.

\frac{3 7 + 4 2 + 3 8 + 4 4 + T}{5} > 41

MultIneq

LHS \cdot 5 > RHS \cdot 5

37 + 42 + 38 + 44 + T > 41 \cdot 5

MultiplyMultiply

37 + 42 + 38 + 44 + T > 205

AddTermsAdd terms

161 + T > 205

SubIneq

LHS - 161 > RHS - 161

T > 44

In order for the bear cubs to come out of their den, the temperature the fifth day must be greater than

4 4^{\circ} F .