__Objective__

At the end of the lesson, students should be able to interpret word problems and represent each information in a linear compound inequality.

There are some problems which are stated in words. Such word problems can be written as algebraic equations or inequalities depending on how the question is put.

**Key Words for Inequalities:**

**At least** – means greater than or equal to.

**No more than, at most** – means less than or equal to

**More than** – means greater than

**Less than** – means less than

Before we look at the examples let’s go over some of the rules and key words for solving word problems.

**Read through the entire problem twice.****Highlight the important information and key words that you need to solve the problem.****Identify your variables.****Write the inequality.****Solve.****Check or justify your answer.**

__Example 1__

The height of a person seeking to get a police job should not be less than 6 ft. Choose an inequality for this information.

a) x ≤ 6

b) x < 6

c) x > 6

d) x ≥ 6

__Solution__

The information says that the person’s height should not be less than 6ft.

Let the person’s height be x.

The inequality that represents the information is x ≥ 6.

[Height of the person should be greater than or equal to 6 ft.]

__Example 2__

The price of a ball should not be more than $2. Write an inequality for this information.

__Solution__

The price of a ball should not be more than $2.

Let the price of ball be x.

The inequality that represents the information is **x ≤ 2.**

[Price of a ball should be less than or equal to $2.]

__Example 3__

The result of doubling a number is at most 20. What is the number?

__Solution __

Let the number be x.

The required inequality is:

Therefore the number is less than or equal to 10.

__Example 4__

The sum of a number and 21 is less than or equal to 37. Find the number.

__Solution__

Let the number be n. **Write the inequality.**

n + 21 ≤ 37 **Subtract 21 from both sides.**

n + 21 – 21 ≤ 37 – 21 **Simplify.**

n ≤ 16

Values of n ≤ 16 satisfy the inequality.

__Example 5__

The sum of a number and 12 is greater than or equal to 57. Find the number.

__Solution__

Let the number be n. **Write the inequality.**

n + 12 ≥ 57 **Subtract 12 from both sides.**

n + 12 – 12 ≥ 57 12 **Simplify.**

n ≥ 45

Values of n ≥ 45 satisfy the inequality.

__Example 6__

One fourth of a number added to one fifth of the same number is less than or equal to 18. Find the range of value of the number.

__Solution __

Let x be the number.

From the question,

**Si****mplify**** the L.H. S.**

M**ultiply both side by the reciprocal of ** **(i .e.****).**

__Compound Inequalities – Word Problems __

__Example 1__

Most snakes must live in places where temperatures range between 75 to 90 degrees. Write an inequality that describes the temp of places where snakes must live.

__Solution__

The important information and key word that you need here is “**temperatures range between 75 to 90 degrees**”.

**Temp >****75˚**** and temp <****90˚**

The two inequalities can be combined into a single inequality as

75˚ < temp < 90˚

The temp of places where snakes must live is **75˚ < temp < 90˚.**

__Example 2__

The heights of Nathan and Ed are 5.8 ft and 6.2 ft. Brian’s height (h) is greater than Nathan’s height but less than Ed’s. Write an inequality that describes the height of Brian.

__Solution__

The height of Brian is greater than Nathan’s height and less than Ed’s height that is **h > 5.8 **and **h < 6.2.**

The two inequalities can be combined into a single inequality as

**5.8 < h < 6.2.**

**So, the compound inequality 5.8 < h < 6.2 shows the height of the Brian**

__Example 3__

The costs of cell phones are either less than $60 or more than $110.

__Solution__

The solution to this compound inequality is all the cost of phone either greater than $1106 or less than $60 either of the inequalities is true. The solution is the combination, or union, of the two individual cost

**Cost < $60 or Cost > $110**

__Example 4__

The audible frequency range of a dog is 25 Hz to 50000 Hz. Write an inequality to describe the frequency range.

__Solution__

The audible frequency range of a dog is from 25 Hz to 50000 Hz that is f is greater than or equal to 25 and less than or equal to 50,000 hertz.

**Express in the form of inequalities:**

**f ≥ 25 **and **f ≤ 50000**

**Combine into a single inequality.**

**25 ≤ f ≤ 50000**

**So, the compound inequality 25 ≤ f ≤ 50000 represents the audible frequency range of a dog.**

Lesson Content

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