<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation

1.9: Inequalities that Describe Patterns

Difficulty Level: Basic Created by: CK-12
Atoms Practice
Estimated8 minsto complete
%
Progress
Practice Inequalities that Describe Patterns
 
 
 
MEMORY METER
This indicates how strong in your memory this concept is
Practice
Progress
Estimated8 minsto complete
%
Estimated8 minsto complete
%
Practice Now
MEMORY METER
This indicates how strong in your memory this concept is
Turn In

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:

  • > “greater than”
  • “greater than or equal to”
  • “less than or equal to”
  • < “less than”
  • “not equal to”

Below are several examples of inequalities:

3x<5x2+2x1>03x4x234x2x

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 a.

1.59(a)<7

Checking a Solution to an Inequality

Unlike equations, inequalities have more than one solution. However, you can check whether a value, such as x=6, 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:

  1. Is m=11 a solution set to 4m+3070?

Plug in m=11, to see if we get a true statement.

4(11)+307044+30707470

Since m=11 gives us a false statement, it is not a solution to the inequality.

  1. Is m=10 a solution to 4m+3070?

Substitute in m=10:

4(10)+307040+30707070

For 7070 to be a true statement, we need 70<70 or 70=70. Since 70=70, this is a true statement, so m=10 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 h.

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

45h<150To 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 h=3 is a solution. Substituting, you get:

45h<15045(3)<150135<150 

This is a true statement since 135 is less than 150. h=3is a solution to this inequality. You could have 3 hours left to travel. 

Example 2

Check whether x=3 is a solution to 2x5<7.

Substitute in x=3, to see if it is a solution to 2x5<7.

2(3)5<765<71<7

Since 1 is less than 7, we have a true statement, so x=3 is a solution to 2x5<7.

Example 3

Check whether x=6 is a solution to 2x5<7.

Check if x=6 is a solution to 2x5<7.

2(6)5<7125<77<7

Since 7 is not less than 7, this is a false statement. Thus x=6 is not a solution to 2x5<7.

Review

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

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

  1. A bus can seat 65 passengers or fewer.
  2. The sum of two consecutive integers is less than 54.
  3. 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.
  4. 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.

  1. x=12; 2(x+6)8x
  2. z=9; 1.4z+5.2>0.4z
  3. y=40; 52y+12<18
  4. t=0.4; 8010(3t+2)

In 12-14, find the solution set.

  1. 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.
  2. Solve the avocado inequality from Example A and check your solution.
  3. 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

  1. Translate into an algebraic equation: 17 less than a number is 65.
  2. Simplify the expression: 34÷(9×3)+62.
  3. Rewrite the following without the multiplication sign: A=12bh.
  4. 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. 

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Please to create your own Highlights / Notes
Show More

Vocabulary

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 \ge greater than or equal to \le less than or equal to < less than \neq not equal to

solution

The value (or multiple values) that make the equation or inequality true.

Image Attributions

Show Hide Details
Description
Difficulty Level:
Basic
Grades:
8 , 9
Date Created:
Feb 24, 2012
Last Modified:
Aug 16, 2016
Save or share your relevant files like activites, homework and worksheet.
To add resources, you must be the owner of the Modality. Click Customize to make your own copy.
Please wait...
Please wait...
Image Detail
Sizes: Medium | Original
 
MAT.ALG.311.L.1
Here