1. 9.4: Inequalities and Absolute Value
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9.4: Inequalities and Absolute Value
Pilar Alcazar
Period 1
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2. ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷× ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×
Equation: |3x+2/4|≤ 5
Since there is a fraction with a denominator of 4, you need to multiply both sides of the equation by 4 or 4/1. Also, the absolute value means you need to do a positive and negative version of the equation.
3x+2/4≤ 5 | 3x+2/4 ≥ -5
x4/1 x4 | x4 x4
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3. ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷× ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×
Equation: |3x+2/4|≤ 5
2. Now, you need to subtract 2 from both sides of the equation because there is a 2 added onto the 3x. Add 2 to both sides of the second equation because the 2 is negative. 3x+2≤ 20 | 3x+2≥ | -2 -2
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4. ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷× ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×
Equation: |3x+2/4|≤ 5
3. Now you want to get the x all by itself. You need to divide both sides by 3 to isolate x. In the second equation, divide both sides by negative three and flip the sign from ≥ to ≤. 3x≤ 18 | 3x≥ -22 ÷3 ÷3 | ÷3 ÷3
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5. ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷× ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×
Equation: |3x+2/4|≤ 5
4. Since x is now isolated, you are finished with the equation. x ≤ 6 | x ≥ -22/3 Answer: {x|-22/3≤x≤6}
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6. ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷× ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×
Critical Thinking: Write an absolute value inequality to describe each of the graphs below.
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7. ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷× ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×
Since the graph only goes to -4 and 2, the inequality would be {x|-4≤x≤2}.
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8. ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷× ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×
Since the graph is less than -2 or greater than 3, the inequality would be {x|x<-2 or x>3}.
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9. ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷× ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×
Equation: |2b-4|< -5
1. Since this is an absolute value equation, you need to write it in a positive and negative form. 2b-4< -5 | 2b-4> 5
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10. ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷× ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×
Equation: |2b-4|< -5
2. Now you need to isolate the number that is multiplied onto b and b itself by either adding or subtracting.
2b-4< | 2b-4> 5 |
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11. ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷× ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×
Equation: |2b-4|< -5
3. Now you need to isolate b by dividing by the number that is multiplied onto it. 2b<-1 | 2b>9 ÷2 ÷2 | ÷2 ÷2
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12. ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷× ÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×÷×
Equation: |2b-4|< -5
Now that b is by itself, you have your answer.
b<-1/ | b>9/2
Answer: {b|b<-1/2 or b>9/2}
If you go back and plug the answer in, you find out that these solutions do not validate. Therefore, there is no solution.
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13. Thanks for Watching!