SOLVING ABSOLUTE VALUE EQUATIONS PRACTICE PROBLEMS

Solve for x :

(1)  10  =  4 + |2y + 1|

(2)  -1  =  -|5x + 1|

(3)  -2|3b - 7| - 9  =  -9

(4)  -3|5x + 1| + 4  =  4

(5)  -2|x + 3|  =  5

(6)  -3|x - 5|  =  7

(7)  0  =  |6x – 9|

(8)  7  =  |4k – 6| + 7

(9)  |-1/5 – 1/2k|  =  9/5

(10)  |-1/6 – 2/9h|  =  1/2

(11)  -3|2 – 6x| + 5  =  -10

(12)  5|1 – 2x| - 7  =  3

(13) |3x – 5|  =  |2x + 1|

(14) |8x + 9|  =  |8x - 1|

(15) |[(2x)/(x + 1)]|  =  5

(16) |[(2x - 1)/(x + 3)]|  =  2

(17) |[(x + 4)/(1 – 2x)]|  =  3/4

1. Answer :

10  =  4 + |2y + 1|

Subtract 4 on both sides, we get

10 - 4  =  4 + |2y + 1| - 4

6  =  |2y + 1|

|2y + 1|  =  6

Now, we have an absolute value equation form.

|2y + 1|  =  6

2y + 1  =  6  (or)  2y + 1  =  -6

2y  =  5  (or)  2y  =  -7

y  =  5/2  (or)  y  =  -7/2

So, the solution of y is 5/2 or -7/2

2. Answer :

-1  =  -|5x + 1|

Dividing by -1 on both sides, we get

1  =  |5x + 1|

|5x + 1|  =  1

Now, we have an absolute value equation form.

5x + 1  =  1  (or)  5x + 1  =  -1

5x  =  0  (or)  5x  =  -2

x  =  0  (or)  x  =  -2/5

So, the solution of x is 0 or -2/5

3. Answer :

-2|3b - 7| - 9  =  -9

Add 9 on both sides, we get

-2|3b - 7| - 9 + 9  =  -9 + 9

-2|3b - 7|  =  0

Dividing by -2 on both sides, we get

|3b - 7|  =  0

Solving for b

3b - 7  =  0

3b  =  7

b  =  7/3 

So, the solution of b is 7/3

4. Answer :

-3|5x + 1| + 4  =  4

Subtract 4 on both sides, we get

-3|5x + 1| + 4 - 4  =  4 - 4

-3|5x + 1|  =  0

Dividing by -3 on both sides, we get

|5x + 1|  =  0

Solving for x

5x + 1  =  0

5x  =  -1

x  =  -1/5 

So, the solution of x is -1/5.

5. Answer :

-2|x + 3|  =  5

Dividing by -2 on both sides, we get

|x + 3|  =  -5/2

Now, we have an absolute value equation that is less than 0.

So, there is no solution for x.

6. Answer :

-3|x - 5|  =  7

Dividing by -3 on both sides, we get

|x - 5|  =  -7/3

Now, we have an absolute value equation that is less than 0

So, there is no solution for x.

7. Answer :

0  =  |6x – 9|

|6x - 9|  =  0

Solving for x

6x – 9  =  0

6x  =  9

x  =  9/6

x  =  3/2

So, the solution of x is 3/2.

8. Answer :

7  =  |4k – 6| + 7

Subtract 7 on both sides, we get

7 - 7  =  |4k – 6| + 7 – 7

0  =  |4k – 6|

|4k - 6|  =  0

Solving for k

4k – 6  =  0

4k  =  6

k  =  6/4

k  =  3/2

So, the solution of k is 3/2.

9. Answer :

|-1/5 – 1/2k|  =  9/5

Solving for k

-1/2k – 1/5  =  9/5  (or)  -1/2k – 1/5  =  -9/5

-1/2k  =  9/5 + 1/5  (or)  -1/2k  =  -9/5 + 1/5

-1/2k  =  2  (or)  -1/2k  =  -8/5

k  =  -4  (or)  k  =  16/5

So, the solution of k is {-4 or 16/5}.

10. Answer :

|-1/6 – 2/9h|  =  1/2

|-2/9h – 1/6|  =  1/2

Solving for h

-2/9h – 1/6  =  1/2  (or)  -2/9h – 1/6  =  -1/2

-2/9h  =  1/2 + 1/6  (or)  -2/9h  =  -1/2 + 1/6

Taking least common multiple,

-2/9h  =  4/6  (or)  -2/9h  =  -2/6

-2/9h  =  2/3  (or)  -2/9h  =  -1/3

h  =  -[(2/3)(9/2)]  (or)  h  =  [(1/3)(9/2)]

h  =  -3  (or)  h  =  3/2

So, the solution of h is {-3 or 3/2}.

11. Answer :

-3|2 – 6x| + 5  =  -10

-3|-6x + 2| + 5  =  -10

Subtract 5 on both sides, we get

-3|-6x + 2| + 5 - 5  =  -10 - 5

-3|-6x + 2|  =  -15

Dividing by -3 on both sides, we get

|-6x + 2|  =  5

Now, we have an absolute value equation form.

Solving for x

-6x + 2  =  5  (or)  -6x + 2  =  -5

-6x  =  3  (or)  -6x  =  -7

x  =  -3/6  (or)  x  =  7/6

x  =  -1/2  (or)  x  =  7/6

So, the solution of x is {-1/2 or 7/6}

12. Answer :

5|1 – 2x| - 7  =  3

5|-2x + 1| - 7  =  3

Add 7 on both sides, we get

5|-2x + 1| - 7 + 7  =  3 + 7

5|-2x + 1|  =  10

Dividing by 5 on both sides, we get

|-2x + 1|  =  2

Solving for x

-2x + 1  =  2  (or)  -2x + 1  =  -2

-2x  =  1  (or)  -2x  =  -3

x  =  -1/2  (or)  x  =  3/2

So, the solution of x is {-1/2 or 3/2}.

13. Answer :

|3x – 5|  =  |2x + 1|

By using absolute value equation property,

3x – 5  =  2x + 1  (or)  3x – 5  =  -(2x + 1)

3x – 2x  =  1 + 5  (or)  3x + 2x  =  -1 + 5

x  =  6  (or)  5x  =  4

x  =  6  (or)  x  =  4/5

So, the solution of x is {6 or 4/5}.

14. Answer :

|8x + 9|  =  |8x - 1|

Solving for x,

8x + 9  =  8x - 1  (or)  8x + 9  =  -(8x – 1)

8x – 8x  =  -1 - 9  (or)  8x + 8x  =  1 - 9

0  =  -10  (or)  16x  =  -8

x  =  -1/2

So, the solution of x is -1/2.

15. Answer :

|[(2x)/(x + 1)]|  =  5

By using absolute value equation property,

[(2x)/(x + 1)]  =  5  (or)  [(2x)/(x + 1)]  =  -5

Solving for x,

2x  =  5(x + 1)  (or)  2x  =  -5(x + 1)

2x  =  5x + 5  (or)  2x  =  -5x – 5

2x – 5x  =  5  (or)  2x + 5x  =  -5

-3x  =  5  (or)  7x  =  -5

x  =  -5/3 (or)  x  =  -5/7

So, the solution of x is {-5/3 or -5/7}.

16. Answer :

|[(2x - 1)/(x + 3)]|  =  2

 [(2x - 1)/(x + 3)]  =  2  (or)  [(2x - 1)/(x + 3)]  =  -2

Solving for x,

2x - 1  =  2(x + 3)  (or)  2x - 1  =  -2(x + 3)

2x  =  2x + 3 + 1  (or)  2x  =  -2x – 6 + 1

By combining like terms,

2x – 2x  =  4  (or)  2x + 2x  =  -5

0  =  4  (or)  4x  =  -5

x  =  -5/4

So, the solution of x is -5/4.

17. Answer :

|[(x + 4)/(1 – 2x)]|  =  3/4

 [(x + 4)/(1 – 2x)]  =  3/4  (or)  [(x + 4)/(1 – 2x)]  =  -3/4

Solving for x,

4(x + 4)  =  3(1 – 2x)  (or)  4(x + 4)  =  -3(1 – 2x)

4x + 16  =  3 – 6x  (or)  4x + 16  =  -3 + 6x

By combining like terms,

4x + 6x  =  3 - 16  (or)  4x - 6x  =  -3 - 16

10x  =  -13  (or)  -2x  =  -19

x  =  -13/10  (or)  x  =  19/2

So, the solution of x is {-13/10 or 19/2}.

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