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Solution - Absolute value equations

Exact form: u=38
u=\frac{3}{8}
Decimal form: u=0.375
u=0.375

Other Ways to Solve

Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
|x|=|y|x=±y and |x|=|y|±x=y
to write all four options of the equation
|4u5|=|4u+2|
without the absolute value bars:

|x|=|y||4u5|=|4u+2|
x=+y(4u5)=(4u+2)
x=y(4u5)=(4u+2)
+x=y(4u5)=(4u+2)
x=y(4u5)=(4u+2)

When simplified, equations x=+y and +x=y are the same and equations x=y and x=y are the same, so we end up with only 2 equations:

|x|=|y||4u5|=|4u+2|
x=+y , +x=y(4u5)=(4u+2)
x=y , x=y(4u5)=(4u+2)

2. Solve the two equations for u

5 additional steps

(4u-5)=(4u+2)

Subtract from both sides:

(4u-5)-4u=(4u+2)-4u

Group like terms:

(4u-4u)-5=(4u+2)-4u

Simplify the arithmetic:

-5=(4u+2)-4u

Group like terms:

-5=(4u-4u)+2

Simplify the arithmetic:

5=2

The statement is false:

5=2

The equation is false so it has no solution.

10 additional steps

(4u-5)=-(4u+2)

Expand the parentheses:

(4u-5)=-4u-2

Add to both sides:

(4u-5)+4u=(-4u-2)+4u

Group like terms:

(4u+4u)-5=(-4u-2)+4u

Simplify the arithmetic:

8u-5=(-4u-2)+4u

Group like terms:

8u-5=(-4u+4u)-2

Simplify the arithmetic:

8u5=2

Add to both sides:

(8u-5)+5=-2+5

Simplify the arithmetic:

8u=2+5

Simplify the arithmetic:

8u=3

Divide both sides by :

(8u)8=38

Simplify the fraction:

u=38

3. Graph

Each line represents the function of one side of the equation:
y=|4u5|
y=|4u+2|
The equation is true where the two lines cross.

Why learn this

We encounter absolute values almost daily. For example: If you walk 3 miles to school, do you also walk minus 3 miles when you go back home? The answer is no because distances use absolute value. The absolute value of the distance between home and school is 3 miles, there or back.
In short, absolute values help us deal with concepts like distance, ranges of possible values, and deviation from a set value.