# 1.4: Equations and Inequalities

**At Grade**Created by: CK-12

## Learning Objectives

- Write equations and inequalities.
- Check solutions to equations.
- Check solutions to inequalities.
- Solve real-world problems using an equation.

## Introduction

In algebra, an **equation** is a mathematical expression that contains an equals sign. It tells us that two expressions represent the same number. For example, \begin{align*}y = 12x\end{align*} is an equation. An **inequality** is a mathematical expression that contains inequality signs. For example, \begin{align*}y \le 12x\end{align*} is an inequality. Inequalities are used to tell us that an expression is either larger or smaller than another expression. Equations and inequalities can contain both **variables** and **constants**.

Variables are usually given a letter and they are used to represent unknown values. These quantities can change because they depend on other numbers in the problem.

Constants are quantities that remain unchanged. Ordinary numbers like \begin{align*}2, \ -3, \ \frac{3}{4},\end{align*} and \begin{align*}\pi\end{align*} are constants.

Equations and inequalities are used as a shorthand notation for situations that involve numerical data. They are very useful because most problems require several steps to arrive at a solution, and it becomes tedious to repeatedly write out the situation in words.

## Write Equations and Inequalities

Here are some examples of equations:

\begin{align*}3x - 2 = 5 \qquad x + 9 = 2x + 5 \qquad \frac{x}{3} = 15 \qquad x^2 + 1 = 10\end{align*}

To write an inequality, we use the following symbols:

> **greater than**

\begin{align*}\ge\end{align*} **greater than or equal to**

< **less than**

\begin{align*}\le\end{align*} **less than or equal to**

\begin{align*}\neq\end{align*} **not equal to**

Here are some examples of inequalities:

\begin{align*}3x < 5 \qquad 4 - x \le 2x \qquad x^2 + 2x - 1 > 0 \qquad \frac{3x}{4} \ge \frac{x}{2} - 3\end{align*}

The most important skill in algebra is the ability to translate a word problem into the correct equation or inequality so you can find the solution easily. The first two steps are **defining the variables** and **translating** the word problem into a mathematical equation.

**Defining the variables** means that we assign letters to any unknown quantities in the problem.

**Translating** means that we change the word expression into a mathematical expression containing variables and mathematical operations with an equal sign or an inequality sign.

**Example 1**

*Define the variables and translate the following expressions into equations.*

a) A number plus 12 is 20.

b) 9 less than twice a number is 33.

c) $20 was one quarter of the money spent on the pizza.

**Solution**

a) **Define**

Let \begin{align*}n=\end{align*} the number we are seeking.

**Translate**

A number plus 12 is 20.

\begin{align*} n + 12 = 20\end{align*}

b) **Define**

Let \begin{align*}n=\end{align*} the number we are seeking.

**Translate**

9 less than twice a number is 33.

This means that twice the number, minus 9, is 33.

\begin{align*}2n - 9 = 33\end{align*}

c) **Define**

Let \begin{align*}m = \end{align*} the money spent on the pizza.

**Translate**

$20 was one quarter of the money spent on the pizza.

\begin{align*}20 = \frac{1}{4} m\end{align*}

Often word problems need to be reworded before you can write an equation.

**Example 2**

*Find the solution to the following problems.*

a) Shyam worked for two hours and packed 24 boxes. How much time did he spend on packing one box?

b) After a 20% discount, a book costs $12. How much was the book before the discount?

**Solution**

a) **Define**

Let \begin{align*}t = \end{align*} time it takes to pack one box.

**Translate**

Shyam worked for two hours and packed 24 boxes. This means that two hours is 24 times the time it takes to pack one box.

\begin{align*}2 = 24t\end{align*}

**Solve**

\begin{align*}t = \frac{2}{24} &= \frac{1}{12} \ \text{hours}\\ \frac{1}{12} \times 60 \ \text{minutes} &= 5 \ \text{minutes} \end{align*}

**Answer**

Shyam takes 5 minutes to pack a box.

b) **Define**

Let \begin{align*}p = \end{align*} the price of the book before the discount.

**Translate**

After a 20% discount, the book costs $12. This means that the price minus 20% of the price is $12.

\begin{align*}p - 0.20p = 12\end{align*}

**Solve**

\begin{align*}p - 0.20p &= 0.8p, \ \text{so} \ 0.8p = 12\\ p &= \frac{12}{0.8} = 15\end{align*}

**Answer**

The price of the book before the discount was $15.

**Check**

If the original price was $15, then the book was discounted by 20% of $15, or $3. \begin{align*}\$15 - 3 = \$12\end{align*}. **The answer checks out**.

**Example 3**

Define the variables and translate the following expressions into inequalities.

a) The sum of 5 and a number is less than or equal to 2.

b) The distance from San Diego to Los Angeles is less than 150 miles.

c) Diego needs to earn more than an 82 on his test to receive a \begin{align*}B\end{align*} in his algebra class.

d) A child needs to be 42 inches or more to go on the roller coaster.

**Solution**

a) **Define**

Let \begin{align*}n =\end{align*} the unknown number.

**Translate**

\begin{align*}5 + n \le 2\end{align*}

b) **Define**

Let \begin{align*}d =\end{align*} the distance from San Diego to Los Angeles in miles.

**Translate**

\begin{align*}d < 150\end{align*}

c) **Define**

Let \begin{align*}x = \end{align*} Diego’s test grade.

**Translate**

\begin{align*}x > 82\end{align*}

d) **Define**

Let \begin{align*}h = \end{align*} the height of child in inches.

**Translate:**

\begin{align*}h \ge 42\end{align*}

## Check Solutions to Equations

You will often need to check solutions to equations in order to check your work. In a math class, checking that you arrived at the correct solution is very good practice. We check the solution to an equation by replacing the variable in an equation with the value of the solution. A solution should result in a true statement when plugged into the equation.

**Example 4**

*Check that the given number is a solution to the corresponding equation.*

a) \begin{align*}y = -1; \ 3y + 5 = -2y\end{align*}

b) \begin{align*}z =3; \ z^2 + 2z = 8\end{align*}

c) \begin{align*}x = -\frac{1}{2}; \ 3x + 1 = x\end{align*}

**Solution**

Replace the variable in each equation with the given value.

a) \begin{align*}3(-1) + 5 & = -2(-1)\\ -3 + 5 & = 2\\ 2 & = 2\end{align*}

**This is a true statement**. This means that \begin{align*}y = -1\end{align*} is a solution to \begin{align*}3y + 5 = -2y\end{align*}.

b) \begin{align*}3^2 + 2(3) & = 8\\ 9 + 6 & = 8\\ 15 & = 8\end{align*}

**This is not a true statement**. This means that \begin{align*}z = 3\end{align*} is **not a solution** to \begin{align*}z^2 + 2z = 8\end{align*} .

c) \begin{align*} 3 \left ( - \frac{1}{2} \right ) + 1 & = - \frac{1}{2}\\ \left ( - \frac{3}{2} \right ) + 1 & = - \frac{1}{2}\\ - \frac{1}{2} & = - \frac{1}{2}\end{align*}

**This is a true statement**. This means that \begin{align*}x = - \frac{1}{2}\end{align*} is a solution to \begin{align*}3x + 1 = x\end{align*}.

## Check Solutions to Inequalities

To check the solution to an inequality, we replace the variable in the inequality with the value of the solution. A solution to an inequality produces a true statement when substituted into the inequality.

**Example 5**

*Check that the given number is a solution to the corresponding inequality.*

a) \begin{align*}a = 10; \ 20a \le 250\end{align*}

b) \begin{align*}b = -0.5; \ \frac{3 - b}{b} > -4\end{align*}

c) \begin{align*}x = \frac{3}{4}; \ 4x + 5 \le 8\end{align*}

**Solution**

Replace the variable in each inequality with the given value.

a) \begin{align*}20(10) & \le 250\\ 200 & \le 250\end{align*}

**This statement is true**. This means that \begin{align*}a = 10\end{align*} is a solution to the inequality \begin{align*}20a \le 250\end{align*}.

Note that \begin{align*}a = 10\end{align*} is not the only solution to this inequality. If we divide both sides of the inequality by 20, we can write it as \begin{align*}a \le 12.5\end{align*}. This means that any number less than or equal to 12.5 is also a solution to the inequality.

b) \begin{align*} \frac{3 - (-0.5)}{(-0.5)} & > -4\\ \frac{3 + 0.5}{-0.5} & > -4\\ - \frac{3.5}{0.5} & > -4\\ -7 & > -4\end{align*}

**This statement is false**. This means that \begin{align*}b = - 0.5\end{align*} is not a solution to the inequality \begin{align*}\frac{3 - b}{b} > -4\end{align*} .

c) \begin{align*}4 \left ( \frac{3}{4} \right ) + 5 & \ge 8\\ 3 + 5 & \ge 8\\ 8 & \ge 8\end{align*}

**This statement is true**. It is true because this inequality includes an equals sign; since 8 is equal to itself, it is also “greater than or equal to” itself. This means that \begin{align*}x = \frac{3}{4}\end{align*} is a solution to the inequality \begin{align*}4x + 5 \le 8\end{align*}.

## Solve Real-World Problems Using an Equation

Let’s use what we have learned about defining variables, writing equations and writing inequalities to solve some real-world problems.

**Example 6**

*Tomatoes cost $0.50 each and avocados cost $2.00 each. Anne buys six more tomatoes than avocados. Her total bill is $8. How many tomatoes and how many avocados did Anne buy?*

**Solution**

**Define**

Let \begin{align*}a = \end{align*} the number of avocados Anne buys.

**Translate**

Anne buys six more tomatoes than avocados. This means that \begin{align*}a + 6 =\end{align*} the number of tomatoes.

Tomatoes cost $0.50 each and avocados cost $2.00 each. Her total bill is $8. This means that .50 times the number of tomatoes plus 2 times the number of avocados equals 8.

\begin{align*}0.5(a + 6) + 2a &= 8\\ 0.5a + 0.5 \cdot 6 + 2a &= 8\\ 2.5a + 3 &= 8\\ 2.5a &= 5\\ a &= 2\end{align*}

Remember that \begin{align*}a =\end{align*} the number of avocados, so Anne buys two avocados. The number of tomatoes is \begin{align*}a + 6 = 2 + 6 = 8\end{align*}.

**Answer**

Anne bought 2 avocados and 8 tomatoes.

**Check**

If Anne bought two avocados and eight tomatoes, the total cost is: \begin{align*}(2 \times 2) + (8 \times 0.5) = 4 + 4 = 8\end{align*}. **The answer checks out.**

**Example 7**

*To organize a picnic Peter needs at least two times as many hamburgers as hot dogs. He has 24 hot dogs. What is the possible number of hamburgers Peter has?*

**Solution**

**Define**

Let \begin{align*}x =\end{align*} number of hamburgers

**Translate**

Peter needs at least two times as many hamburgers as hot dogs. He has 24 hot dogs.

This means that twice the number of hot dogs is less than or equal to the number of hamburgers.

\begin{align*}2 \times 24 \le x, \ \text{or} \ 48 \le x\end{align*}

**Answer**

Peter needs at least 48 hamburgers.

**Check**

48 hamburgers is twice the number of hot dogs. So more than 48 hamburgers is more than twice the number of hot dogs. **The answer checks out**.

## Additional Resources

For more practice solving inequalities, check out http://www.aaastudy.com/equ725x7.htm.

## Review Questions

- Define the variables and translate the following expressions into equations.
- Peter’s Lawn Mowing Service charges $10 per job and $0.20 per square yard. Peter earns $25 for a job.
- Renting the ice-skating rink for a birthday party costs $200 plus $4 per person. The rental costs $324 in total.
- Renting a car costs $55 per day plus $0.45 per mile. The cost of the rental is $100.
- Nadia gave Peter 4 more blocks than he already had. He already had 7 blocks.

- Define the variables and translate the following expressions into inequalities.
- A bus can seat 65 passengers or fewer.
- The sum of two consecutive integers is less than 54.
- The product of a number and 3 is greater than 30.
- An amount of money is invested at 5% annual interest. The interest earned at the end of the year is greater than or equal to $250.
- You buy hamburgers at a fast food restaurant. A hamburger costs $0.49. You have at most $3 to spend. Write an inequality for the number of hamburgers you can buy.
- Mariel needs at least 7 extra credit points to improve her grade in English class. Additional book reports are worth 2 extra credit points each. Write an inequality for the number of book reports Mariel needs to do.

- Check whether the given number is a solution to the corresponding equation.
- \begin{align*}a = -3; \ 4a + 3 = -9\end{align*}
- \begin{align*}x = \frac{4}{3}; \ \frac{3}{4}x + \frac{1}{2} = \frac{3}{2}\end{align*}
- \begin{align*}y = 2; \ 2.5y - 10.0 = -5.0\end{align*}
- \begin{align*}z = -5; \ 2(5 - 2z) = 20 - 2(z - 1)\end{align*}

- Check whether the given number is a solution to the corresponding inequality.
- \begin{align*}x = 12; \ 2(x + 6) \le 8x\end{align*}
- \begin{align*}z = -9; \ 1.4z + 5.2 > 0.4z\end{align*}
- \begin{align*}y = 40; \ - \frac{5}{2}y + \frac{1}{2} < -18\end{align*}
- \begin{align*}t = 0.4; \ 80 \ge 10(3t + 2)\end{align*}

- The cost of a Ford Focus is 27% of the price of a Lexus GS 450h. If the price of the Ford is $15000, what is the price of the Lexus?
- On your new job you can be paid in one of two ways. You can either be paid $1000 per month plus 6% commission of total sales or be paid $1200 per month plus 5% commission on sales over $2000. For what amount of sales is the first option better than the second option? Assume there are always sales over $2000.
- A phone company offers a choice of three text-messaging plans. Plan A gives you unlimited text messages for $10 a month; Plan B gives you 60 text messages for $5 a month and then charges you $0.05 for each additional message; and Plan C has no monthly fee but charges you $0.10 per message.
- If \begin{align*}m\end{align*} is the number of messages you send per month, write an expression for the monthly cost of each of the three plans.
- For what values of \begin{align*}m\end{align*} is Plan A cheaper than Plan B?
- For what values of \begin{align*}m\end{align*} is Plan A cheaper than Plan C?
- For what values of \begin{align*}m\end{align*} is Plan B cheaper than Plan C?
- For what values of \begin{align*}m\end{align*} is Plan A the cheapest of all? (Hint: for what values is A both cheaper than B and cheaper than C?)
- For what values of \begin{align*}m\end{align*} is Plan B the cheapest of all? (Careful—for what values is B cheaper than A?)
- For what values of \begin{align*}m\end{align*} is Plan C the cheapest of all?
- If you send 30 messages per month, which plan is cheapest?
- What is the cost of each of the three plans if you send 30 messages per month?

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