# 1.4: Equations and Inequalities

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

When an algebraic expression is set equal to another value, variable, or expression, a new mathematical sentence is created. This sentence is called an **equation.**

**Definition:** An **algebraic equation** is a mathematical sentence connecting an expression to a value, a variable, or another expression with an equal sign (=).

Consider the theme park situation from lesson 1.3. Suppose there is a concession stand selling burgers and French fries. Each burger costs $2.50 and each order of French fries costs $1.75. You and your family will spend exactly $25.00 on food. How many burgers can be purchased? How many orders of fries? How many of each type can be purchased if your family plans to buy a combination of burgers and fries?

The underlined word exactly lends a clue to the type of mathematical sentence you will need to write to model this situation.

These words can be used to symbolize the equal sign:

*Exactly, equivalent, the same as, identical, is*

The word *exactly* is synonymous with equal, so this word is directing us to write an equation. Using the methods learned in lessons 1.2 and 1.3, read every word in the sentence and translate each into mathematical symbols.

**Example 1:** Your family is planning to purchase only burgers. How many can be purchased with $25.00?

**Solution:**

Step 1: Choose a variable to represent the unknown quantity, say \begin{align*}b\end{align*} for burgers.

Step 2: Write an equation to represent the situation: \begin{align*}2.50 b = 25.00\end{align*}.

Step 3: Think. What number multiplied by 2.50 equals 25.00?

The solution is 10, so your family can purchase exactly ten burgers.

**Example 2:** Translate the following into equations:

a) *9 less than twice a number is* 33.

b) *Five more than four times a number is* 21.

c) $20.00 *was one-quarter of the money spent on pizza.*

**Solutions:**

a) Let “a number” be \begin{align*}n\end{align*}. So, twice a number is \begin{align*}2n\end{align*}.

Nine less than that is \begin{align*}2n - 9\end{align*}.

The word *is* means the equal sign, so \begin{align*}2n - 9 = 33\end{align*}.

b) Let “a number” be \begin{align*}x\end{align*}. So five more than four times a number is 21 can be written as: \begin{align*}4x + 5 = 21\end{align*}.

c) Let “of the money” be \begin{align*}m\end{align*}. The equation could be written as \begin{align*}\frac{1}{4} m = 20.00\end{align*}.

**Definition:** The **solution** to an equation or inequality is the value (or multiple values) that make the equation or inequality true.

Using statement (c) from example 2, find the solution.

\begin{align*}\frac{1}{4} m = 20.00\end{align*}

Think: One-quarter can also be thought of as *divide by four.* What divided by 4 equals 20.00?

The solution is 80. So, the money spent on pizza was $80.00.

Checking an answer to an equation is almost as important as the equation itself. By substituting the value for the variable, you are making sure both sides of the equation balance.

**Example 3:** Check that \begin{align*}x = 5\end{align*} is the solution to the equation \begin{align*}3x + 2 = -2x + 27\end{align*}.

**Solution:** To check that \begin{align*}x = 5\end{align*} is the solution to the equation, substitute the value of 5 for the variable, \begin{align*}x\end{align*}:

\begin{align*}3x + 2 &= -2x + 27\\ 3 \cdot x + 2 &= -2 \cdot x + 27\\ 3 \cdot 5 + 2 &= -2 \cdot 5 + 27\\ 15 + 2 &= -10 + 27\\ 17 &= 17\end{align*}

Because \begin{align*}17 = 17\end{align*} is a true statement, we can conclude that \begin{align*}x = 5\end{align*} is a solution to \begin{align*}3x + 2 = -2x + 27\end{align*}.

**Example 4:** Is \begin{align*}z = 3\end{align*} a solution to \begin{align*}z^2 + 2z = 8\end{align*}?

**Solution:** Begin by substituting the value of 3 for \begin{align*}z\end{align*}.

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

Because \begin{align*}15 = 8\end{align*} is NOT a true statement, we can conclude that \begin{align*}z = 3\end{align*} is not a solution to \begin{align*}z^2 + 2z = 8\end{align*}.

## Sometimes Things Are Not Equal

In some cases there are multiple answers to a problem or the situation requires something that is not exactly equal to another value. When a mathematical sentence involves something other than an equal sign, an **inequality** is formed.

**Definition:** An **algebraic inequality** is a mathematical sentence connecting an expression to a value, a variable, or another expression with an inequality sign.

Listed below are the most common inequality signs.

\begin{align*} > \end{align*} “greater than”

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

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

\begin{align*}<\end{align*} “less than”

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

Below are several examples of inequalities.

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

**Example 5:** Translate the following into an inequality: Avocados cost $1.59 per pound. How many pounds of avocados can be purchased for less than $7.00?

**Solution:** Choose a variable to represent the number of pounds of avocados purchased, say \begin{align*}a\end{align*}.

\begin{align*}1.59(a)<7\end{align*}

You will be asked to solve this inequality in the exercises

## Checking the Solution to an Inequality

Unlike equations, inequalities typically have more than one solution. Checking solutions to inequalities is more complex than checking solutions to equations. The key to checking a solution to an inequality is to choose a number that occurs within the solution set.

**Example 6:** Check that \begin{align*}m \le 10\end{align*} is a solution to \begin{align*}4m + 30 \le 70\end{align*}.

**Solution:** If the solution set is true, any value less than or equal to 10 should make the original inequality true.

Choose a value less than 10, say 4. Substitute this value for the variable \begin{align*}m\end{align*}.

\begin{align*}&4(4) + 30\\ &16 + 30\\ &46 \le 70\end{align*}

The value found when \begin{align*}m = 4\end{align*} is less than 70. Therefore, the solution set is true.

Why was the value 10 not chosen? Endpoints are not chosen when checking an inequality because the direction of the inequality needs to be tested. Special care needs to be taken when checking the solutions to an inequality.

## 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. CK-12 Basic Algebra: Equations and Inequalities (16:11)

- Define
*solution*. - What is the difference between an algebraic equation and an algebraic inequality? Give an example of each.
- What are the five most common inequality symbols?

In 4 – 11, define the variables and translate the following statements into algebraic 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.
- A bus can seat 65 passengers or fewer.
- The sum of two consecutive integers is less than 54.
- 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.

In 12 – 15, check that 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*}

For exercises 16 – 19, check that 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*}

In 20 – 24, find the value of the variable.

- \begin{align*}m + 3 = 10\end{align*}
- \begin{align*}6 \times k = 96\end{align*}
- \begin{align*}9 - f = 1\end{align*}
- \begin{align*}8h = 808\end{align*}
- \begin{align*}a + 348 = 0\end{align*}
- Using the burger and French fries situation from the lesson, give three combinations of burgers and fries your family can buy without spending more than $25.00.
- Solve the avocado inequality from Example 5 and check your solution.
- You are having a party and are making sliders. Each person will eat 5 sliders. There will be seven people at your party. How many sliders do you need to make?
- The cost of a Ford Focus is 27% of the price of a Lexus GS 450h. If the price of the Ford is $15,000, 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 on 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.
- Suppose your family will purchase only orders of French fries using the information found in the opener of this lesson. How many orders of fries can be purchased for $25.00?

**Mixed Review**

- Translate into an algebraic equation: 17 less than a number is 65.
- Simplify the expression: \begin{align*}3^4 \div (9 \times 3)+6-2\end{align*}.
- Rewrite the following without the multiplication sign: \begin{align*}A = \frac{1}{2} \cdot b \cdot h\end{align*}.
- The volume of a box without a lid is given by the formula \begin{align*}V = 4x (10-x)^2\end{align*}, where \begin{align*}x\end{align*} is a length in inches and \begin{align*}V\end{align*} is the volume in cubic inches. What is the volume of the box when \begin{align*}x=2\end{align*}?

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