Lesson Absolute Value equations

Algebra ->  Absolute-value -> Lesson Absolute Value equations      Log On


   

This Lesson (Absolute Value equations) was created by by ikleyn(52777) About Me : View Source, Show
About ikleyn:

Absolute Value equations


In this lesson you can learn how to solve simple equations with the unknown under the Absolute Value sign.

Problem 1

Solve absolute value equation  abs%28x%29 = 2.

Solution

By the definition,  abs%28x%29 = x  if  x >= 0,  and  abs%28x%29 = -x  if  x < 0.
Therefore, we should try to solve two equations.

The first equation is  x = 2  at  x >= 0.
The second equation is  -x = 2  at  x < 0.

The first equation has the solution  x = 2,  and it satisfies the condition  x >= 0.
The second equation has the solution  x = -2,  and it satisfies the condition  x < 0.

So, the given equation has two solutions:  x = 2  and  x = -2.

You can check the solutions by substituting the found values into the given equation:
1)  abs%282%29 = 2,  and
2)  abs%28-2%29 = 2.


Problem 2

Solve absolute value equation  abs%28x%29 = 5.5.

Solution

By the definition,  abs%28x%29 = x  if  x >= 0,  and  abs%28x%29 = -x  if  x < 0.
Therefore, we should try to solve two equations.

The first equation is  x = 5.5  at  x >= 0.
The second equation is  -x = 5.5  at  x < 0.

The first equation has the solution  x = 5.5,  and it satisfies the condition  x >= 0.
The second equation has the solution  x = -5.5,  and it satisfies the condition  x < 0.

So, the given equation has two solutions:  x = 5.5  and  x = -5.5.

You can check the solutions by substituting the found values into the given equation:
1)  abs%285.5%29 = 5.5,  and
2)  abs%28-5.5%29 = 5.5.


Problem 3

Solve absolute value equation  abs%28x%29%2B4 = 5.

Solution

Let us simplify the equation first by distracting 4 from both sides:  abs%28x%29 = 1.

Next, by the definition,  abs%28x%29 = x  if  x >= 0,  and  abs%28x%29 = -x  if  x < 0.
Therefore, we should try to solve two equations.

The first equation is  x = 1  at  x >= 0.
The second equation is  -x = 1  at  x < 0.

The first equation has the solution  x = 1,  and it satisfies the condition  x >= 0.
The second equation has the solution  x = -1,  and it satisfies the condition  x < 0.

So, the given equation has two solutions:  x = 1  and  x = -1.

You can check the solutions by substituting the found values into the given equation:
1)  abs%281%29%2B4 = 1%2B4 = 5,  and
2)  abs%28-1%29+%2B4%29 = 1%2B4 = 5.


Problem 4

Solve absolute value equation  abs%28x%2B2%29 = 7.

Solution

By the definition,  abs%28x%2B2%29 = x%2B2  if  x%2B2 >= 0,  and  abs%28x%2B2%29 = -%28x%2B2%29  if  x%2B2 < 0.
Therefore, we should try to solve two equations.

The first equation is  x%2B2 = 7  at  x >= -2.
The second equation is  -%28x%2B2%29 = 7  at  x < -2.

The first equation has the solution  x = 5,  and it satisfies the condition  x >= -2.
The second equation has the solution  x = -9,  and it satisfies the condition  x < -2.

So, the given equation has two solutions:  x = 5  and  x = -9.

You can check the solutions by substituting the found values into the given equation:
1)  abs%285%2B2%29 = abs%287%29 = 7,  and
2)  abs%28-9%2B2%29 = abs%28-7%29 = 7.


Problem 5

Solve absolute value equation  abs%282x%2B5%29 = 9.

Solution

By the definition,  abs%282x%2B5%29 = 2x%2B5  if  2x%2B5 >= 0,  and  abs%282x%2B5%29 = -%282x%2B5%29  if  2x%2B5 < 0.
Therefore, we should try to solve two equations.

The first equation is  2x%2B5 = 9  at  x >= -5%2F2.
The second equation is  -%282x%2B5%29 = 9  at  x < -5%2F2.

The first equation has the solution  x = 2,  and it satisfies the condition  x >= -5%2F2.
The second equation has the solution  x = -7,  and it satisfies the condition  x < -5%2F2.

So, the given equation has two solutions:  x = 2  and  x = -7.

You can check the solutions by substituting the found values into the given equation:
1)  abs%282%2A2%2B5%29 = abs%289%29 = 9,  and
2)  abs%282%2A%28-7%29%2B5%29 = abs%28-14%2B5%29 = abs%28-9%29 = 9.


For more complicated equations with the unknown under the Absolute Value sign see the lessons
    - HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 1,
    - HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 2,
    - HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 3,
    - HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 4

    - HOW TO solve equations containing Quadratic Terms under the Absolute Value sign. Lesson 1
    - HOW TO solve equations containing Quadratic Terms under the Absolute Value sign. Lesson 2
under the current topic in this site.

See also the lesson  OVERVIEW of lessons on Absolute Value equations  under the current topic in this site.

Use this file/link  ALGEBRA-I - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-I.

This lesson has been accessed 3924 times.