Answer :
Sure, let's solve each absolute value equation step-by-step:
10. Solve [tex]\( |x+6| = 13 \)[/tex]
When solving an equation involving absolute value, we set up two separate equations:
1. [tex]\( x + 6 = 13 \)[/tex]
- Subtract 6 from both sides:
- [tex]\( x = 13 - 6 \)[/tex]
- [tex]\( x = 7 \)[/tex]
2. [tex]\( x + 6 = -13 \)[/tex]
- Subtract 6 from both sides:
- [tex]\( x = -13 - 6 \)[/tex]
- [tex]\( x = -19 \)[/tex]
So, the solution set is [tex]\( x = 7 \)[/tex] and [tex]\( x = -19 \)[/tex].
11. Solve [tex]\( |x-11| + 19 = 5 \)[/tex]
First, we need to isolate the absolute value:
- Subtract 19 from both sides:
- [tex]\( |x - 11| = 5 - 19 \)[/tex]
- [tex]\( |x - 11| = -14 \)[/tex]
An absolute value cannot equal a negative number, so this equation has no solution.
12. Solve [tex]\( 4|x+2| - 2 = 18 \)[/tex]
First, isolate the absolute value:
- Add 2 to both sides:
- [tex]\( 4|x+2| = 20 \)[/tex]
- Divide both sides by 4:
- [tex]\( |x+2| = 5 \)[/tex]
Now, set up two equations for the absolute value:
1. [tex]\( x + 2 = 5 \)[/tex]
- Subtract 2 from both sides:
- [tex]\( x = 5 - 2 \)[/tex]
- [tex]\( x = 3 \)[/tex]
2. [tex]\( x + 2 = -5 \)[/tex]
- Subtract 2 from both sides:
- [tex]\( x = -5 - 2 \)[/tex]
- [tex]\( x = -7 \)[/tex]
So, the solution set is [tex]\( x = 3 \)[/tex] and [tex]\( x = -7 \)[/tex].
13. Solve [tex]\( 16.4 + |x-5| = 27.4 \)[/tex]
First, isolate the absolute value:
- Subtract 16.4 from both sides:
- [tex]\( |x-5| = 27.4 - 16.4 \)[/tex]
- [tex]\( |x-5| = 11 \)[/tex]
Now, set up two equations for the absolute value:
1. [tex]\( x - 5 = 11 \)[/tex]
- Add 5 to both sides:
- [tex]\( x = 11 + 5 \)[/tex]
- [tex]\( x = 16 \)[/tex]
2. [tex]\( x - 5 = -11 \)[/tex]
- Add 5 to both sides:
- [tex]\( x = -11 + 5 \)[/tex]
- [tex]\( x = -6 \)[/tex]
So, the solution set is [tex]\( x = 16 \)[/tex] and [tex]\( x = -6 \)[/tex].
Matching with the solution sets given:
- 10. Solution: [tex]\( x = 7 \)[/tex] and [tex]\( x = -19 \)[/tex] — Matches answer C.
- 11. Solution: No solution — Matches answer D.
- 12. Solution: [tex]\( x = 3 \)[/tex] and [tex]\( x = -7 \)[/tex] — Matches answer A.
- 13. Solution: [tex]\( x = 16 \)[/tex] and [tex]\( x = -6 \)[/tex] — Matches answer B.
10. Solve [tex]\( |x+6| = 13 \)[/tex]
When solving an equation involving absolute value, we set up two separate equations:
1. [tex]\( x + 6 = 13 \)[/tex]
- Subtract 6 from both sides:
- [tex]\( x = 13 - 6 \)[/tex]
- [tex]\( x = 7 \)[/tex]
2. [tex]\( x + 6 = -13 \)[/tex]
- Subtract 6 from both sides:
- [tex]\( x = -13 - 6 \)[/tex]
- [tex]\( x = -19 \)[/tex]
So, the solution set is [tex]\( x = 7 \)[/tex] and [tex]\( x = -19 \)[/tex].
11. Solve [tex]\( |x-11| + 19 = 5 \)[/tex]
First, we need to isolate the absolute value:
- Subtract 19 from both sides:
- [tex]\( |x - 11| = 5 - 19 \)[/tex]
- [tex]\( |x - 11| = -14 \)[/tex]
An absolute value cannot equal a negative number, so this equation has no solution.
12. Solve [tex]\( 4|x+2| - 2 = 18 \)[/tex]
First, isolate the absolute value:
- Add 2 to both sides:
- [tex]\( 4|x+2| = 20 \)[/tex]
- Divide both sides by 4:
- [tex]\( |x+2| = 5 \)[/tex]
Now, set up two equations for the absolute value:
1. [tex]\( x + 2 = 5 \)[/tex]
- Subtract 2 from both sides:
- [tex]\( x = 5 - 2 \)[/tex]
- [tex]\( x = 3 \)[/tex]
2. [tex]\( x + 2 = -5 \)[/tex]
- Subtract 2 from both sides:
- [tex]\( x = -5 - 2 \)[/tex]
- [tex]\( x = -7 \)[/tex]
So, the solution set is [tex]\( x = 3 \)[/tex] and [tex]\( x = -7 \)[/tex].
13. Solve [tex]\( 16.4 + |x-5| = 27.4 \)[/tex]
First, isolate the absolute value:
- Subtract 16.4 from both sides:
- [tex]\( |x-5| = 27.4 - 16.4 \)[/tex]
- [tex]\( |x-5| = 11 \)[/tex]
Now, set up two equations for the absolute value:
1. [tex]\( x - 5 = 11 \)[/tex]
- Add 5 to both sides:
- [tex]\( x = 11 + 5 \)[/tex]
- [tex]\( x = 16 \)[/tex]
2. [tex]\( x - 5 = -11 \)[/tex]
- Add 5 to both sides:
- [tex]\( x = -11 + 5 \)[/tex]
- [tex]\( x = -6 \)[/tex]
So, the solution set is [tex]\( x = 16 \)[/tex] and [tex]\( x = -6 \)[/tex].
Matching with the solution sets given:
- 10. Solution: [tex]\( x = 7 \)[/tex] and [tex]\( x = -19 \)[/tex] — Matches answer C.
- 11. Solution: No solution — Matches answer D.
- 12. Solution: [tex]\( x = 3 \)[/tex] and [tex]\( x = -7 \)[/tex] — Matches answer A.
- 13. Solution: [tex]\( x = 16 \)[/tex] and [tex]\( x = -6 \)[/tex] — Matches answer B.