Consider that you are driving a car at 45 miles per hour and you know that your destination is less than 150 miles away. What inequality could you set up to solve for the number of hours that you have left to travel? After you've solved the inequality, how could you check to make sure that your answer is correct?

### Inequalities

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.

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*}

#### Let's translate the following statement into an inequality:

Avocados cost $1.59 per pound. How many pounds of avocados can be purchased for less than $7.00?

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*}

**Checking a Solution to an Inequality**

Unlike equations, inequalities have more than one solution. However, you can check whether a value, such as \begin{align*}x=6\end{align*}, is *a* solution to an inequality the same way as you would check if it is the solution to an equation--by substituting it in and seeing if you get a true algebraic statement.

#### Now, let's check the solution for the following inequalities:

- Is \begin{align*}m = 11\end{align*} a solution set to \begin{align*}4m + 30 \le 70\end{align*}?

Plug in \begin{align*}m = 11\end{align*}, to see if we get a true statement.

\begin{align*}&4(11) + 30\le 70\\ &44 + 30\le 70\\ &74 \le 70\end{align*}

Since \begin{align*}m = 11\end{align*} gives us a false statement, it is not a solution to the inequality.

- Is \begin{align*}m = 10\end{align*} a solution to \begin{align*}4m + 30 \le 70\end{align*}?

Substitute in \begin{align*}m = 10\end{align*}:

\begin{align*}&4(10) + 30\le 70\\ &40 + 30\le 70\\ &70 \le 70\end{align*}

For \begin{align*}70 \le 70\end{align*} to be a true statement, we need \begin{align*}70<70\end{align*} or \begin{align*}70=70\end{align*}. Since \begin{align*}70=70\end{align*}, this is a true statement, so \begin{align*}m = 10\end{align*} is a solution.

### Examples

#### Example 1

Earlier, you were told that you are driving a car at 45 miles per hour and your destination is less than 150 miles away. What inequality could you set up to solve for the number of hours that you have left to travel? How can you check to make sure that your answer is correct?

Choose a variable to represent the number of hours left to travel, say \begin{align*}h\end{align*}.

If you travel 45 miles per hour for \begin{align*}h\end{align*} hours, then the expression \begin{align*}45h\end{align*} represents the total number of miles you traveled. This value is less than 150 so the inequality that represents the situation is:

\begin{align*}45h<150\end{align*}To check a solution to this equation, you would substitute in the value and make sure that the statement is valid. For example, let's check that \begin{align*}h=3\end{align*} is a solution. Substituting, you get:

\begin{align*}45h < 150\\ 45(3) < 150\\ 135 < 150\end{align*}

This is a true statement since 135 is less than 150. \begin{align*}h=3\end{align*}is a solution to this inequality. You could have 3 hours left to travel.

#### Example 2

Check whether \begin{align*}x=3 \end{align*} is a solution to \begin{align*}2x-5<7.\end{align*}

Substitute in \begin{align*}x=3 \end{align*}, to see if it is a solution to \begin{align*}2x-5<7.\end{align*}

\begin{align*}2(3)-5< 7\\ 6-5<7\\ 1<7 \end{align*}

Since 1 is less than 7, we have a true statement, so \begin{align*}x=3 \end{align*} is a solution to \begin{align*}2x-5<7.\end{align*}

#### Example 3

Check whether \begin{align*}x=6 \end{align*} is a solution to \begin{align*}2x-5<7.\end{align*}

Check if \begin{align*}x=6 \end{align*} is a solution to \begin{align*}2x-5<7.\end{align*}

\begin{align*}2(6)-5<7\\ 12-5<7\\ 7<7 \end{align*}

Since 7 is not less than 7, this is a false statement. Thus \begin{align*}x=6 \end{align*} is not a solution to \begin{align*}2x-5<7.\end{align*}

### Review

- 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 4–7, define the variables and translate the following statements into algebraic equations.

- 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.

For exercises 8–11, check whether the given solution set is the solution set 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 12-14, find the solution set.

- Using the burger and French fries situation from the previous Concept, give three combinations of burgers and fries your family can buy without spending more than $25.00.
- Solve the avocado inequality from Example A and check your solution.
- 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.

**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*}?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.9.