ALI - Creating Linear Inequalities to Solve Problems (Lesson)

Create Linear Inequalities to Solve Problems

We use inequalities in math to represent a range of numbers restricted by some sort of constraint. Let's review solving some basic inequalities:

Two examples of inequalities:

1.

Notice on the second example: the sign changed when we divided by a negative value. Anytime you multiply or divide by a negative, you must flip the sign.

5x−2>18            Add 2 on both sides.

5x>20                    Divide by 5 on both sides.

x>4

2.

3x−7≤5x+11     Subtract 5x on both sides.

−2x−7≤11            Add 7 on both sides.

−2x≤18                    Divide by -2 on both sides.

x≥−9

Let's look at these symbols and some common words associated with them:

Symbol

Common Phrases

Examples
Less Than <

less than

x is less than 4

x<4
Greater Than >

higher than, must exceed

the sum of a number and 3 must exceed 7

x+3>7
Less Than or Equal To  <

at most, not higher than, maximum, greatest number

twice a number is at most 93

2x≤93
Greater Than or Equal To  >

at least, minimum, is not less than, smallest number

you must score at least a 70

x≥70

 

We've been writing and solving linear equations, so now let's try some inequalities. But first, let's think about some common steps we can use for inequalities or equations:

  1. Draw a picture, if applicable
  2. Define your variable
  3. Set up your equation
  4. Solve your equation
  5. Check to be sure you answered the question!

Example: The sum of two consecutive integers is less than 83. Find the pair of integers with the greatest sum.

  • No picture to draw
  • Let x = the first consecutive number, so x + 1 = the second consecutive number
  • Equation:  x + x + 1 < 83
  • Solve: 2x + 1 < 83 (subtract 1 from both sides)
    • 2x < 82 (divide by 2 on both sides)
    • x < 41 
  • So the first of the numbers must be an integer less than 41, so x = 40, which means x + 1 = 41. Check to be sure these two numbers add up to less than 83: 40 + 41 = 81 which is less than 83!

Watch this video to try a few more:

Write and Solve Linear Inequalities Practice

Now try these problems to be sure you have it down!  Make sure that as you practice, you are constructing your answers in complete sentences.

  1. The sum of twice a number and 7 is less than 27. Write and solve an inequality statement to figure out the constraints on this number.
  2. Your quiz grades are 78, 72, 87, and 90. What score on the fifth quiz will make your average quiz grade at least 82?               
  3. Solve the inequality: −3x+7>5−x                                                       
  4. Solve the inequality: −2x+3≤5
  5. You have the option between two phone plans. Option A charges $25 a month plus $0.10 per MB of data used, or Option B that charges $10 a month plus $0.20 per MB of data used. How much data do you have to use to make Option A the better option?

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

IMAGES CREATED BY GAVS