(Greater than)
1. Undo the operations outside absolute value bars, so that the problem looks like this.
2. Split the above problem into two inequalities similar to the diagram below. Solve for the unknown in each equation. (Notice that the absolute value bars have been dropped.)
3. Graph the resulting solution set.
Directions: Please solve for the unknown in the following absolute value inequality.
4|3x-5|+1>17
Step 1. Notice that the absolute value quantity is not isolated on one side of the inequality. We must subtract one on both sides then divide by four on both sides to rectify this.
4|3x-5|>16
Step 2. Now that the problem is consistent with the diagram above, we can split the problem into two inequalities and solve them. Notice in the inequality on the right, along with making the number negative, we flipped the inequality symbol.
|3x-5|>4 becomes
3x-5>4 Or 3x-5<-4
3x>9 Or 3x<1
x>3 Or x<1/3
Step 3. The graph must contain the numbers that are smaller than or equal to negative third and greater than or equal to positive three. Note that the less than or equal to symbol and the greater than or equal to symbol require open dots when they are graphed.
Directions: Please solve for the unknown in the following absolute value inequalities. If there is no solution, please indicate so by stating "null set". Also, graph the solutions sets.
1. |3x|>9 2. |18b|>9 3. |2x+12|>14
4. |10c|-15>15 5. 9|6r|>-63 6. 9|4-2y|-6>30
7. -5|6+2r|>20 8. 7|7x|-7>42 9. 3|4+2g|>21
10. 4-|3b-8|<-8
Directions: Please write 1-2 paragraphs that do the following.
1. Compare and contrast the steps used to solve absolute value inequalities with less than to the steps used to solve absolute value inequalities with greater than. You must remark on the similarities and the differences between the diagrams provided on these web pages. Are the methods more similar to each other or different than each other? Can you describe a more general process that works for solving both types of inequalities?