When we solve absolute value functions with two absolute signs, we follow the rules given below.
More simply,
|u| = |v|
u = v and -u = v
Solve the following absolute value equations.
Problem 1 :
|x + 4| = |2x - 7|
Solution :
|x + 4| = |2x - 7|
x + 4 = 2x - 7 x - 2x = -7 - 4 -x = - 11 x = 11 |
x + 4 = - (2x - 7) x + 4 = -2x + 7 x + 2x = 7 - 4 3x = 3 x = 1 |
So, the solution is x = 1 or x = 11.
Problem 2 :
|3x + 5| = |x - 6|
Solution :
|3x + 5| = |x - 6|
3x + 5 = x - 6 3x - x = -6 - 5 2x = -11 x = -11/2 |
3x + 5 = -(x - 6) 3x + 5 = -x + 6 3x + x = 6 - 5 4x = 1 x = 1/4 |
So, the solution is x = -11/2 or x = 1/4.
Problem 3 :
|x - 9| = |x + 6|
Solution :
|x - 9| = |x + 6|
x - 9 = x + 6 x - x = 6 - 9 0 = -3 it is not true. |
|x - 9| = |x + 6| x - 9 = - (x + 6) x - 9 = -x - 6 x + x = -6 + 9 2x = 3 x = 3/2 |
So, the solution is 3/2.
Problem 4:
|x + 4| = |x - 3|
Solution :
|x + 4| = |x - 3|
x + 4 = x - 3 x - x = -3 - 4 0 = -7 it is not true. |
|x + 4| = |x - 3| x + 4 = -(x - 3) x + 4 = -x + 3 x + x = 3 - 4 2x = -1 x = -1/2 |
So, the solution is -1/2.
Problem 5 :
|5t + 7| = |4t + 3|
Solution :
|5t + 7| = |4t + 3|
5t + 7 = 4t + 3 5t - 4t = 3 - 7 t = -4 |
|5t + 7| = |4t + 3| 5t + 7 = - (4t + 3) 5t + 7 = -4t - 3 5t + 4t = -3 - 7 9t = -10 t = -10/9 |
o, the solution is t = -4 or t = -10/9.
Problem 6 :
|3a - 1| = |2a + 4|
Solution :
|3a - 1| = |2a + 4|
3a - 1 = 2a + 4 3a - 2a = 4 + 1 a = 5 |
|3a - 1| = |2a + 4| 3a - 1 = - (2a + 4) 3a - 1 = -2a - 4 3a + 2a = -4 + 1 5a = -3 a = -3/5 |
So, the solution is a = 5 or a = -3/5.
Problem 7 :
|n - 3| = |3 - n|
Solution :
|n - 3| = |3 - n|
True for all real values of n. So, infinitely many solution.
Problem 8 :
|y - 2| = |2 - y|
Solution :
|y - 2| = |2 - y|
True for all real values of y. So, infinitely many solution.
Problem 9 :
|7 - a| = |a + 5|
Solution :
|7 - a| = |a + 5|
7 - a = a + 5 -a - a = 5 - 7 -2a = -2 a = 1 |
|7 - a| = |a + 5| 7 - a = - (a + 5) 7 - a = -a - 5 -a + a = -5 - 7 0 = -12 |
So, the solution is a = 1.
Problem 10 :
|6 - t| = |t + 7|
Solution :
|6 - t| = |t + 7|
6 - t = t + 7 -t - t = 7 - 6 -2t = 1 t = -1/2 |
|6 - t| = |t + 7| 6 - t = - (t + 7) 6 - t = -t - 7 -t + t = -7 - 6 0 = -13 |
So, the solution is t = -1/2.
Problem 11 :
|1/2x - 5| = |1/4x + 3|
Solution :
|1/2x - 5| = |1/4x + 3| 1/2x - 5 = 1/4x + 3 1/2x - 1/4x = 3 + 5 (2x - x)/4 = 8 x/4 = 8 x = 32 |
|1/2x - 5| = |1/4x + 3| 1/2x - 5 = - (1/4x + 3) 1/2x - 5 = -1/4x - 3 1/2x + 1/4x = -3 + 5 (2x + x)/4 = 2 3x/4 = 2 3x = 8 x = 8/3 |
So, the solution is x = 32 or x = 8/3.
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