1.9: Inequalities that Describe Patterns
What if you were driving a car at 45 miles per hour and you knew that your destination was 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? Once you've completed this Concept, you'll be able to find and verify solutions to inequalities representing scenarios like these.
Guidance
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*}
\begin{align*}\ge\end{align*}
\begin{align*}\le\end{align*}
\begin{align*}<\end{align*}
\begin{align*}\neq\end{align*}
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 A
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 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*}
The following two examples show you how this works.
Example B
Check whether \begin{align*}m = 11\end{align*}
Solution:
Plug in \begin{align*}m = 11\end{align*}
\begin{align*}&4(11) + 30\le 70\\
&44 + 30\le 70\\
&74 \le 70\end{align*}
Since \begin{align*}m = 11\end{align*}
Example C
Check whether \begin{align*}m = 10\end{align*}
Solution:
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*}
Guided Practice
1. Check whether \begin{align*}x=3 \end{align*}
2. Check whether \begin{align*}x=6 \end{align*}
Solutions:
1. Substitute in \begin{align*}x=3 \end{align*}
\begin{align*}2(3)5< 7\\
65<7\\
1<7 \end{align*}
Since 1 is less than 7, we have a true statement, so \begin{align*}x=3 \end{align*}
2. Check if \begin{align*}x=6 \end{align*}
\begin{align*}2(6)5<7\\
125<7\\
7<7
\end{align*}
Since 7 is not less than 7, this is a false statement. Thus \begin{align*}x=6 \end{align*}
Practice
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. CK12 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 – 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*}
x=12; 2(x+6)≤8x 
\begin{align*}z = 9; \ 1.4z + 5.2 > 0.4z\end{align*}
z=−9; 1.4z+5.2>0.4z 
\begin{align*}y = 40; \ \frac{5}{2}y + \frac{1}{2} <  18\end{align*}
y=40; −52y+12<−18 
\begin{align*}t = 0.4; \ 80 \ge 10(3t + 2)\end{align*}
t=0.4; 80≥10(3t+2)
In 1214, 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)+62\end{align*}
34÷(9×3)+6−2 .  Rewrite the following without the multiplication sign: \begin{align*}A = \frac{1}{2} \cdot b \cdot h\end{align*}
A=12⋅b⋅h .  The volume of a box without a lid is given by the formula \begin{align*}V = 4x (10x)^2\end{align*}
V=4x(10−x)2 , where \begin{align*}x\end{align*}x is a length in inches and \begin{align*}V\end{align*}V is the volume in cubic inches. What is the volume of the box when \begin{align*}x=2\end{align*}x=2 ?
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algebraic inequality
An algebraic inequality is a mathematical sentence connecting an expression to a value, a variable, or another expression with an inequality sign.inequality signs
Listed below are the most common inequality signs. greater than greater than or equal to less than or equal to less than not equal tosolution
The value (or multiple values) that make the equation or inequality true.Image Attributions
Here you will learn how to read about a reallife situation and write an inequality that represents this situation. You will then solve the inequality and plug the answer back into the inequality to check your work.