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Solution - Absolute value equations

Exact form: x=-9,-511
x=-9 , -\frac{5}{11}
Decimal form: x=9,0.455
x=-9 , -0.455

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
|4x11|=|7x+16|
without the absolute value bars:

|x|=|y||4x11|=|7x+16|
x=+y(4x11)=(7x+16)
x=y(4x11)=(7x+16)
+x=y(4x11)=(7x+16)
x=y(4x11)=(7x+16)

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||4x11|=|7x+16|
x=+y , +x=y(4x11)=(7x+16)
x=y , x=y(4x11)=(7x+16)

2. Solve the two equations for x

13 additional steps

(4x-11)=(7x+16)

Subtract from both sides:

(4x-11)-7x=(7x+16)-7x

Group like terms:

(4x-7x)-11=(7x+16)-7x

Simplify the arithmetic:

-3x-11=(7x+16)-7x

Group like terms:

-3x-11=(7x-7x)+16

Simplify the arithmetic:

3x11=16

Add to both sides:

(-3x-11)+11=16+11

Simplify the arithmetic:

3x=16+11

Simplify the arithmetic:

3x=27

Divide both sides by :

(-3x)-3=27-3

Cancel out the negatives:

3x3=27-3

Simplify the fraction:

x=27-3

Move the negative sign from the denominator to the numerator:

x=-273

Find the greatest common factor of the numerator and denominator:

x=(-9·3)(1·3)

Factor out and cancel the greatest common factor:

x=9

10 additional steps

(4x-11)=-(7x+16)

Expand the parentheses:

(4x-11)=-7x-16

Add to both sides:

(4x-11)+7x=(-7x-16)+7x

Group like terms:

(4x+7x)-11=(-7x-16)+7x

Simplify the arithmetic:

11x-11=(-7x-16)+7x

Group like terms:

11x-11=(-7x+7x)-16

Simplify the arithmetic:

11x11=16

Add to both sides:

(11x-11)+11=-16+11

Simplify the arithmetic:

11x=16+11

Simplify the arithmetic:

11x=5

Divide both sides by :

(11x)11=-511

Simplify the fraction:

x=-511

3. List the solutions

x=-9,-511
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|4x11|
y=|7x+16|
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.