Equations and Inequalities: True and False Statements

Contributor: Erika Wargo. Lesson ID: 12893

What does an alligator have to do with mathematics? It can help you remember when to use > and < and other math symbols. Learn how number sentences and variables are used to express inequalities!

categories

Middle School

subject
Math
learning style
Visual
personality style
Otter
Grade Level
Middle School (6-8)
Lesson Type
Quick Query

Lesson Plan - Get It!

Audio:

The new ride at the fair has height requirements.

The minimum height is 36 inches and the maximum height is 72 inches. If you are 62 inches tall, can you go on the ride? What would be a height that would not be permitted to ride?

Did you know that sometimes in math you don’t have to find the exact answer?

Most of the time, you are trying to solve an equation that has one answer. An equation includes an equal sign (=) and represents two expressions that have the same value. But sometimes, two expressions are greater than or less than each other and are represented using an inequality. Inequalities tell about expressions that have different values.

Watch a video to learn about inequalities. Respond to the following questions in your math journal:

  1. Draw the symbol that represents each comparison: equal, not equal to, greater than, less than, greater than or equal to, less than or equal to.
  2. Why are inequalities used?
  3. How can inequalities be used to represent real-world problems?

Discuss the questions above with a parent or teacher after watching the Math Is Fun video clip (ends at 5:40), Equal, Greater or Less Than:

 

As you saw in the video above, there are special signs used with inequalities:

equal to =
not equal to
less than <
less than or equal to
greater than >
greater than or equal to

 

When using the symbols above, be sure the wide part of the sign is facing towards the greater value and the small part points to the smaller number. For example, to show that 15 is greater than 12, you would write 15 > 12.

One way to remember this is to think of the sign as an alligator’s mouth, and the alligator always wants to eat the larger side of the inequality!

An equation or inequality can be true or false, depending on the symbol used. In some problems, you will need to substitute a value for the variable and solve it. In other problems, you might have to find a value that will result in a true or false number sentence. A number sentence is a mathematical sentence — usually an equation or inequality — written with numbers and mathematical symbols.

Example Substitute 3 for x in each equation or inequality and evaluate. Determine if it results in a true number sentence or a false number sentence:

  1. 6 + x = 19
    • Substitute 3 for x, so 6 + 3 =9. 9 ≠19, so this number sentence is false.
  2. 4x = 12
    • Substitute 3 for x, so 4(3) = 12. This problem means to multiply, and 4 x 3 = 12, so this number sentence is true.
  3. 8 + x ≤ 12
    • Substitute 3 for x, so 8 + 3 = 11. 11 ≤ 12 means that 11 is less than or equal to 12, which is true. The inequality sign states that the expression on the left side has to be less than or equal to the right side.
  4. Find a value for x to make this number sentence false: 39 - x ≥ 26.
    • First, find a number that would make the inequality true by being equal to 26. Since 39 - 13 = 26, if x = 13, the left side of the expression equals the right side.
    • The left side of the expression also needs to be greater than 26 to be true, so that would include numbers less than 13. If you want the expression to be false, you would need to choose numbers that would make the left side of the equation less than 26. In this example, x would need to be a number that is greater than or equal to 14 to be false.

Example At the beginning of the lesson, you were presented with this word problem:

The new ride at the fair has height requirements. The minimum height is 36 inches and the maximum height is 72 inches. If you are 62 inches tall, can you go on the ride? What would be a height that would not be permitted to ride?

The minimum height is 36 inches, which means that is the smallest value that is allowed to ride. The maximum height is 72 inches, which means that is the greatest value that is allowed to ride. If "h" stands for height, the inequalities would be h ≥ 36 and h ≤ 72. The rider must be equal to or greater than 36 inches, but less than or equal to 72 inches. If you are 62 inches, that is less than 72 inches, so you would be able to go on the ride. If you were 32 inches or 75 inches, you would not be able to go on the ride since your height is less than 36 inches or greater than 72 inches.

In your math journal, write a response to the following questions:

  • Describe the difference between these symbols: < and ≤.
  • How do you determine if a number sentence is true or false?

Now, you will move on to the Got It? section to complete interactive practice to determine if equations and inequality statements are true or false.

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