Question

let f(x)=-|x - 4|+7, then: 11. solve f(x)=5. 12. solve f(x)=3. 13. solve f(x)=7. solve each equation algebraically. check your answers and show all work: 14. |5x| = 20 15. 2|x - 5|+3 = 11 16. -3|x + 1|=9 use the graph below of the absolute - value function f(x) to answer #16 and #17. 17. what is the solution set to f(x)>4? 18. what is the solution set to f(x)≤4?

Explanation:

11. Solve \(f(x)=5\) where \(f(x)= -|x - 4|+7\)

Step1: Substitute \(f(x)\)

Set \(-|x - 4|+7 = 5\).

Step2: Isolate the absolute - value term

Subtract 7 from both sides: \(-|x - 4|=5 - 7=- 2\). Then multiply both sides by - 1 to get \(|x - 4| = 2\).

Step3: Solve the absolute - value equation

We have two cases: \(x-4 = 2\) or \(x - 4=-2\).
For \(x-4 = 2\), add 4 to both sides: \(x=2 + 4=6\).
For \(x - 4=-2\), add 4 to both sides: \(x=-2 + 4 = 2\).

12. Solve \(f(x)=3\) where \(f(x)= -|x - 4|+7\)

Step1: Substitute \(f(x)\)

Set \(-|x - 4|+7 = 3\).

Step2: Isolate the absolute - value term

Subtract 7 from both sides: \(-|x - 4|=3 - 7=-4\). Multiply both sides by - 1 to get \(|x - 4| = 4\).

Step3: Solve the absolute - value equation

We have two cases: \(x-4 = 4\) or \(x - 4=-4\).
For \(x-4 = 4\), add 4 to both sides: \(x=4 + 4=8\).
For \(x - 4=-4\), add 4 to both sides: \(x=-4 + 4 = 0\).

13. Solve \(f(x)=7\) where \(f(x)= -|x - 4|+7\)

Step1: Substitute \(f(x)\)

Set \(-|x - 4|+7 = 7\).

Step2: Isolate the absolute - value term

Subtract 7 from both sides: \(-|x - 4|=7 - 7 = 0\). Then \(|x - 4| = 0\).

Step3: Solve the absolute - value equation

If \(|x - 4| = 0\), then \(x-4 = 0\), so \(x = 4\).

14. Solve \(|5x| = 20\)

Step1: Consider two cases

Case 1: \(5x=20\), then \(x=\frac{20}{5}=4\).
Case 2: \(5x=-20\), then \(x=\frac{-20}{5}=-4\).

15. Solve \(2|x - 5|+3 = 11\)

Step1: Isolate the absolute - value term

Subtract 3 from both sides: \(2|x - 5|=11 - 3 = 8\). Then divide both sides by 2: \(|x - 5| = 4\).

Step2: Solve the absolute - value equation

We have two cases: \(x-5 = 4\) or \(x - 5=-4\).
For \(x-5 = 4\), add 5 to both sides: \(x=4 + 5=9\).
For \(x - 5=-4\), add 5 to both sides: \(x=-4 + 5 = 1\).

16. Solve \(-3|x + 1|=9\)

Step1: Isolate the absolute - value term

Divide both sides by - 3: \(|x + 1|=-3\). Since the absolute - value of a number is non - negative, there is no solution.

Answer:

1. \(x = 2\) or \(x = 6\)
2. \(x = 0\) or \(x = 8\)
3. \(x = 4\)
4. \(x = 4\) or \(x=-4\)
5. \(x = 1\) or \(x = 9\)
6. No solution