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