Solve the following absolute value equations :
1) |3x + 5| = 7
2) |7x| = 21
3) |2x + 5| + 6 = 7
4) |x - 3| + 6 = 6
5) 2|3x +4| = 7
6) 3|5x - 6| - 4 = 5
7) |x2 - 4x - 5| = 7
8) 0.5|0.5x| - 0.5 = 2.5
9) If the absolute value equation |2x + k| = 3 has the solution x = -2, find the value of k.
10) If the absolute value equation |x - 3| - k = 0 has the solution x = -5, find the value of k.
1. Answer :
|3x + 5| = 7
3x + 5 = 7 or 3x + 5 = -7
3x = 2 or 3x = -12
x = 2/3 or x = -4
2. Answer :
|7x| = 21
7x = 21 or 7x = -21
x = 3 or x = -3
3. Answer :
|2x + 5| + 6 = 7
Subtract 6 from each side.
|2x + 5| = 1
2x + 5 = 1 or 2x + 5 = 1
2x = -4 or 2x = -6
x = -2 or x = -3
4. Answer :
|x - 3| + 6 = 6
Subtract 6 from each side.
|x - 3| = 0
x - 3 = 0
x = 3
5. Answer :
2|3x +4| = 7
Divide each side by 2.
|3x + 4| = 7/2
3x + 4 = 7/2 or 3x + 4 = -7/2
3x = 7/2 - 4 or 3x = -7/2 - 4
3x = -1/2 or 3x = -15/2
x = -1/6 or x = -15/6
x = -1/6 or x = -5/2
6. Answer :
3|5x - 6| - 4 = 5
Add 4 to each side.
3|5x - 6| = 9
Divide each side by 3.
|5x - 6| = 3
5x - 6 = 3 or 5x - 6 = -3
5x = 9 or 5x = 3
x = 9/5 or x = 3/5
7. Answer :
|x2 - 4x - 5| = 7
x2 - 4x - 5 = 7 or x2 - 4x - 5 = -7
x2 - 4x - 12 = 0 or x2 - 4x + 2 = 0
Solve the first quadratic equation x2 - 4x - 12 = 0.
x2 - 4x - 12 = 0
(x + 2)(x - 6) = 0
x + 2 = 0 or x - 6 = 0
x = -2 or x = 6
Solve the second quadratic equation x2 - 4x + 2 = 0.
This quadratic equation can not be solved by factoring. So, we can use quadratic formula and solve the equation as shown below.
Comparing ax2 + bx + c = 0 and x2 - 4x + 2 = 0, we get
a = 1, b = -4, c = 2
Quadratic formula :
Substitute a = 1, b = -4 and c = 2.
Therefore x = -2 or 6 or 2 - √2 or 2 + √2.
8. Answer :
0.5|0.5x| - 0.5 = 2.5
Add 0.5 to each side.
0.5|0.5x| = 3
Divide each side by 0.5
|0.5x| = 6
0.5x = 6 or 0.5x = -6
x = 12 or x = -12
9. Answer :
Because x = -2 is a solution, substitute x = -2 in the given absolute value equation.
|2(-2) + k| = 3
|-4 + k| = 3
Solve for k :
-4 + k = 3 or -4 + k = -3
k = 7 or k = 1
10. Answer :
Because x = -2 is a solution, substitute x = -2 in the given absolute value equation.
|-5 - 3| - k = 0
|-8| - k = 0
8 - k = 0
8 = k
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