Janet holds up a card that reads . Donna holds up a card that reads . Andrew says they are not the same but Donna argues with him. Show, using an example, that Andrew is correct.

### Watch This

Khan Academy One Step Inequalities

### Guidance

One variable linear inequalities have a different form than one variable linear equations. Linear equations have the general form of , where . Linear inequalities can have one of four forms: or . You should notice the difference is that instead of an equals sign, there is an inequality symbol.

When you solve for a linear inequality, you follow the same rules as you would for a linear equation; however, you must remember one big rule: *If you divide or multiply by a negative number while solving, you must reverse the sign of the inequality.*

#### Example A

In the following table, a linear equation has been solved. Solve for the inequality using the similar steps. Are the steps the same? Is the inequality still true if you substitute 8 in for ?

Equation |
Inequality |
Is the inequality still true? |
---|---|---|

? | ||

**Solution:**

Equation |
Inequality |
Is the inequality still true? |
---|---|---|

no | ||

No, there is no difference in the steps used to find the two solutions.

#### Example B

In the following table, a linear equation has been solved. Solve for the inequality using the similar steps. Are the steps the same? Is the inequality still true if you substitute 6 in for ?

Equation |
Inequality |
Is the inequality still true? |
---|---|---|

? | ||

**Solution:**

Equation |
Inequality |
Is the inequality still true? |
---|---|---|

yes | ||

No, there is no difference in the steps used to find the two solutions.

#### Example C

In the following table, a linear equation has been solved. Solve for the inequality using the similar steps. Are the steps the same? Is the inequality still true if you substitute 3 in for ?

Equation |
Inequality |
Is the inequality still true? |
---|---|---|

? | ||

**Solution:**

Equation |
Inequality |
Is the inequality still true? |
---|---|---|

yes | ||

Yes, there was a difference in the steps used for the two solutions. When dividing by –3, the sign of the inequality was reversed.

#### Concept Problem Revisited

Janet holds up a card that reads . Donna holds up a card that reads . Andrew says they are not the same but Donna argues with him. Show, using an example, that Andrew is correct.

Andrew could use a real world example. For example, say Andrew held out two $5 bills and six $1 bills. Andrew holds Janet’s card and says, "Is this true?"

The answer would be yes.

Now let’s try it with Donna's inequality.

This amount of money is not greater than $16; it is just equal to $16. The two mathematical statements are not the same.

### Vocabulary

- Linear Inequality
can have one of four forms: , or .*Linear inequalities*

### Guided Practice

1. In the following table, a linear equation has been solved. Solve for the inequality using similar steps, but remember if you multiply or divide by a negative number you should reverse the inequality sign. Is the inequality still true if you substitute –10 in for ?

Equation |
Inequality |
Is the inequality still true? |
---|---|---|

? | ||

2. In the following table, a linear equation has been solved. Solve for the inequality using similar steps, but remember if you multiply or divide by a negative number you should reverse the inequality sign. Is the inequality still true if you substitute 6 in for ?

Equation |
Inequality |
Is the inequality still true? |
---|---|---|

? | ||

3. In the following table, a linear equation has been solved. Solve for the inequality using similar steps, but remember if you multiply or divide by a negative number you should reverse the inequality sign. Is the inequality still true if you substitute –10 in for ?

Equation |
Inequality |
Is the inequality still true? |
---|---|---|

? | ||

**Answers:**

1.

Equation |
Inequality |
Is the inequality still true? |
---|---|---|

no | ||

2.

Equation |
Inequality |
Is the inequality still true? |
---|---|---|

no | ||

3.

Equation |
Inequality |
Is the inequality still true? |
---|---|---|

yes | ||

### Practice

In the following table, a linear equation has been solved.

Equation |
Inequality |
Is the inequality still true? |
---|---|---|

? | ||

- Solve for the inequality using similar steps.
- Were all of the steps the same? Why or why not?
- Is the inequality still true if you substitute –4.5 in for ?

In the following table, a linear equation has been solved.

Equation |
Inequality |
Is the inequality still true? |
---|---|---|

? | ||

- Solve for the inequality using similar steps.
- Were all of the steps the same? Why or why not?
- Is the inequality still true if you substitute 8 in for ?

In the following table, a linear equation has been solved.

Equation |
Inequality |
Is the inequality still true? |
---|---|---|

? | ||

- Solve for the inequality using similar steps.
- Were all of the steps the same? Why or why not?
- Is the inequality still true if you substitute –2 in for ?

- The sum of two numbers is greater than 764. If one of the numbers is 416, what could the other number be?
- 205 less a number is greater than or equal to 112. What could that number be?
- Five more than twice a number is less than 20. If the number is a whole number, what could the number be?
- The product of 7 and a number is greater than 42. If the number is a whole number less than 10, what could the number be?
- Three less than 5 times a number is less than or equal to 12. If the number is a whole number, what could the number be?
- Double a number and add 12 and the result will be greater than 20. The number is less than 6. What is the number?

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 2.9.