2.9: One Variable Inequalities
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 inequalities have a different form than one variable 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, the sign of the inequality is reversed.
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? Answer the question if the inequality is still true if you substitute 8 in for ?
Equation | Inequality | Is the inequality still true? |
---|---|---|
? | ||
Is there any difference between the rules used in the two solutions?
Equation | Inequality | Is the inequality still true? |
---|---|---|
no | ||
No, there is no difference in the rules used for 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? Answer the question if the inequality is still true if you substitute 6 in for ?
Equation | Inequality | Is the inequality still true? |
---|---|---|
? | ||
Equation | Inequality | Is the inequality still true? |
---|---|---|
yes | ||
No, there is no difference in the rules used for 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? Answer the question if the inequality is still true if you substitute 3 in for ?
Equation | Inequality | Is the inequality still true? |
---|---|---|
? | ||
Equation | Inequality | Is the inequality still true? |
---|---|---|
yes | ||
Yes, there was a difference in the rules 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 help 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 Donnas’ inequality.
This amount of money is not greater than $16; it is just equal to $16. Andrew then proved the two are not equal.
Vocabulary
- Linear Inequality
- Linear inequalities can have one of four forms: , or . In other words, the left side no longer equals the right side, it is less than, greater than, less than or equal to, or greater than or equal to.
Guided Practice
1. In the following table, a linear equation has been solved. Solve for the inequality using the similar steps. Are the steps the same? Answer the question if the inequality is 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 the similar steps. Are the steps the same? Answer the question if the inequality is 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 the similar steps. Are the steps the same? Answer the question if the inequality is 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 | ||
Yes, there was a difference in the rules used for the two solutions. When dividing by -3, the sign of the inequality was reversed.
2.
Equation | Inequality | Is the inequality still true? |
---|---|---|
no | ||
Yes, there was a difference in the rules used for the two solutions. When dividing by -3, the sign of the inequality was reversed.
3.
Equation | Inequality | Is the inequality still true? |
---|---|---|
yes | ||
Practice
- In the following table, a linear equation has been solved. Solve for the inequality using the similar steps. Are the steps the same? Answer the question if the inequality is still true if you substitute -4.5 in for ?
Equation | Inequality | Is the inequality still true? |
---|---|---|
? | ||
- In the following table, a linear equation has been solved. Solve for the inequality using the similar steps. Are the steps the same? Answer the question if the inequality is still true if you substitute 8 in for ?
Equation | Inequality | Is the inequality still true? |
---|---|---|
? | ||
- In the following table, a linear equation has been solved. Solve for the inequality using the similar steps. Are the steps the same? Answer the question if the inequality is still true if you substitute -2 in for ?
Equation | Inequality | Is the inequality still true? |
---|---|---|
? | ||
- The sum of two numbers is 764. If one of the numbers is 416, what could the solution be?
- Two hundred and five less a number is greater than or equal to one hundred and twelve. What could that solution be?
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Learning Objectives
Here you are going to learn about one variable inequalities.
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Date Created:
Dec 19, 2012Last Modified:
Jan 30, 2013Vocabulary
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