(Less than)
Step 1. Undo the operations outside absolute value bars, so that the problem looks like the diagram below.
Step 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.)
Step 3. Graph the resulting solution set.
Directions: Please solve for the unknown in the following absolute value inequality.
4|x-4|-1<15
Step 1. Notice that the absolute value quantity is not isolated on one side of the inequality. We must add one on both sides then divide by four on both sides to rectify this.
4|x-4|<16
|x-4|<4
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.
|x-4|<4 becomes
x-4<4 AND x-4>-4
x<8 AND x>0
Step 3. The graph must contain the numbers that are larger than negative eight and less than positive eight. Note that the strictly less than symbol and the strictly greater than 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. |2y-7|<9 2. |18b|<9 3. |2x+12|<14
4. |10c|-15<15 5. |-2c+4|<10 6. 9|4+2y|+12<30
7. 4|6-2x|<20 8. 7|7x|-7<42 9. 3|4+2g|<21
10. 5|-6r|<-45
Directions: Please write 1-2 paragraphs that thoroughly address the following.
1. Compare and contrast the level of difficulty of problem 1 and problem 6 above. Which is harder? Which has more steps and why? Is there an underlying rule that applies to both problems at some point while solving both problems?