# ABSOLUTE VALUE EQUATIONS WITH TWO ABSOLUTE VALUES

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 - 7x - 2x = -7 - 4-x = - 11x = 11 x + 4 = - (2x - 7)x + 4 = -2x + 7x + 2x = 7 - 43x = 3x = 1

So, the solution is x = 1 or x = 11.

Problem 2 :

|3x + 5| = |x - 6|

Solution :

|3x + 5| = |x - 6|

 3x + 5 = x - 63x - x = -6 - 52x = -11x = -11/2 3x + 5 = -(x - 6)3x + 5 = -x + 63x + x = 6 - 54x = 1x = 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 + 6x - x = 6 - 90 = -3it is not true. |x - 9| = |x + 6|x - 9 = - (x + 6)x - 9 = -x - 6x + x = -6 + 92x = 3x = 3/2

So, the solution is 3/2.

Problem 4:

|x + 4| = |x - 3|

Solution :

|x + 4| = |x - 3|

 x + 4 = x - 3x - x = -3 - 40 = -7it is not true. |x + 4| = |x - 3|x + 4 = -(x - 3)x + 4 = -x + 3x + x = 3 - 42x = -1x = -1/2

So, the solution is -1/2.

Problem 5 :

|5t + 7| = |4t + 3|

Solution :

|5t + 7| = |4t + 3|

 5t + 7 = 4t + 35t - 4t = 3 - 7t = -4 |5t + 7| = |4t + 3|5t + 7 = - (4t + 3)5t + 7 = -4t - 35t + 4t = -3 - 79t = -10t = -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 + 43a - 2a = 4 + 1a = 5 |3a - 1| = |2a + 4|3a - 1 = - (2a + 4)3a - 1 = -2a - 43a + 2a = -4 + 15a = -3a = -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 = -2a = 1 |7 - a| = |a + 5|7 - a = - (a + 5)7 - a = -a - 5-a + a = -5 - 70 = -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 = 1t = -1/2 |6 - t| = |t + 7|6 - t = - (t + 7)6 - t = -t - 7-t + t = -7 - 60 = -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 + 31/2x - 1/4x = 3 + 5(2x - x)/4 = 8x/4 = 8x = 32 |1/2x - 5| = |1/4x + 3|1/2x - 5 = - (1/4x + 3)1/2x - 5 = -1/4x - 31/2x + 1/4x = -3 + 5(2x + x)/4 = 23x/4 = 23x = 8x = 8/3

So, the solution is x = 32 or x = 8/3.

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