# INTEGERS AND ABSOLUTE VALUE WORKSHEET

## About "Integers and absolute value worksheet"

Integers and absolute value worksheet is much useful to the kids who would like to practice problems on absolute value and absolute value equations.

Before we look at integers and absolute value worksheet, let us look at some basic stuff about solving absolute value equations.

"Solving absolute value equations" is the stuff which is being studied by the students who study high school math.

Here we are going to see, "How to solve an absolute value equation"

And do remember, before solving any absolute value function, it has to be in the form of |x + a| = k (There should be only absolute part on the left side)

(Here "a" and "k" are real numbers)

Let us consider the absolute value equation |2x + 3| = 5.

The picture given below clearly explains, "How the above absolute value equation can be solved".

## Integer and absolute value worksheet - Problems

1)  Solve the absolute value function |3x + 5| = 7

2)  Solve the absolute value function |7x| = 21

3)  Solve the absolute value function |2x + 5| + 6  = 7

4)  Solve the absolute value function |x - 3| + 6  = 6

5)  Solve the absolute value function 2|3x +4|  = 7

6)  Solve the absolute value function 3|5x - 6|- 4  = 5

7)  Solve the absolute value function |x² - 4x - 5| = 7

8)  Solve the absolute value function 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".

## Integer and absolute value worksheet - Answers

Problem 1 :

Solve the absolute value function |3x + 5| = 7

Solution :

When we apply the method explained above for |3x + 5| = 7,

we get  3x + 5 = 7   or    3x + 5 = -7

3x = 2    or     3x = -12

x = 2/3     or     x =   -4

Hence the solution is x = -4, 2/3

Let us look at the next problem on "Integers and absolute value worksheet".

Problem 2 :

Solve the absolute value function |7x| = 21

Solution :

When we apply the method explained above for |7x| = 21,

we get  7x = 21   or    7x = -21

x = 3    or     x = -3

Hence the solution is x = -3, 3.

Let us look at the next problem on "Integers and absolute value worksheet".

Problem 3 :

Solve the absolute value function |2x + 5| + 6  = 7

Solution :

Let us write the given absolute value equation in the form

|x+a| = k

|2x + 5| + 6  = 7 ------------> |2x+5| = 1

When we apply the method explained above for |2x + 5| = 1,

we get  2x + 5 = 1   or    2x + 5 = -1

2x = -4    or     2x = -6

x = -2     or     x = -3

Hence the solution is x = -2, -3.

Let us look at the next problem on "Integers and absolute value worksheet"

Problem 4 :

Solve the absolute value function |x - 3| + 6  = 6

Solution :

Let us write the given absolute value equation in the form

|x+a| = k

|x - 3| + 6  = 6 ------------> |x - 3| = 0

When we apply the method explained above for |x - 3| = 0,

we get  x -3 = 0   or    x -3 = 0

x = 3  or   x = 3

Hence the solution is x = 3, 3.

Let us look at the next problem on "Integers and absolute value worksheet"

Problem 5 :

Solve the absolute value function 2|3x +4|  = 7

Solution :

Let us write the given absolute value equation in the form

|x+a| = k

2|3x+4| = 7 -------> |3x+4| = 7/2

When we apply the method explained above for |3x + 4| = 7/2,

we get  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 = -5/2

Hence the solution is x = -1/6, -5/2

Let us look at the next problem on "Integers and absolute value worksheet"

Problem 6 :

Solve the absolute value function 3|5x - 6|- 4  = 5

Solution :

Let us write the given absolute value equation in the form

|x+a| = k

3|5x-6|-4 = 5 -------> 3|5x-6| = 9 ------->|5x-6| = 3

When we apply the method explained above for |5x - 6| = 3,

we get  5x - 6 = 3   or    5x -6  = -3

5x = 9    or     5x = 3

x = 9/5     or     x = 3/5

Hence the solution is x = 9/5, 3/5

Now, let us look at a problem on "Integers and absolute value" with quadratic polynomial in absolute sign.

Problem 7 :

Solve the absolute value function |x² - 4x - 5| = 7

Solution :

When we apply the method explained above for |x²-4x-5| = 7,

we get  x² - 4x  - 5 = 7   or    x² -4x - 5 = -7

x² - 4x - 12 = 0    or     x² - 4x + 2  = 0

Let us 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

Let us solve the second quadratic equation x² - 4x + 2 = 0

This quadratic equation can not be solved using factoring. Because the left side part can not be factored.

So, we can use quadratic formula and solve the equation as given below.

Let us look at the next problem on "Integers and absolute value worksheet"

Problem 8 :

Solve the absolute value function 0.5|0.5x| - 0.5 = 2.5

Solution :

Let us write the given absolute value equation in the form

|x+a| = k

0.5|0.5x|-0.5=2.5 ------> 0.5|0.5x|=3 ------> |0.5x| = 6

When we apply the method explained above for |0.5x| = 6,

we get  0.5x  = 6   or    0.5x = -6

x = 12     or     x = -12

Hence the solution is x = -12, 12

Now let us look at some quiet different problems on "Integers and absolute value"

Problem 9 :

If the absolute value equation |2x+k| = 3 has the solution x= -2, find the value of "k".

Solution :

Since x = -2 is a solution, we can plug x = -3 in the given absolute value equation |2x+k| = 3.

|2(-2)+k| = 3 -------|-4+k| = 3

Using the method explained above, we get

-4 + k = 3          or        -4 + k = -3

k = 7                  or                k = 1

Hence, the value of k = 1, 7

Let us look at the next problem on "Integers and absolute value worksheet"

Problem 10 :

If the absolute value equation |x - 3| - k = 0 has the solution x = -5, find the value of "k".

Solution :

Let us write the given absolute value equation in the form

|x+a| = k

|x-3|-k --------> |x-3| = k

Since x = 5 is a solution, we can plug x = 5 in the absolute value equation |x-3| = k

|-5-3| = k

|-8| = k

8 = k     (absolute value of any negative number is positive)

Hence the value of k = 8.

After having gone through the step by step solutions for all the problems on "Integers and absolute value worksheet",  we hope that the students would have understood how to do problems on "Integers and absolute value worksheet".

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