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Solution - Absolute value equations

Exact form: c=11,13
c=11 , \frac{1}{3}
Decimal form: c=11,0.333
c=11 , 0.333

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
|2c6|=|c+5|
without the absolute value bars:

|x|=|y||2c6|=|c+5|
x=+y(2c6)=(c+5)
x=y(2c6)=(c+5)
+x=y(2c6)=(c+5)
x=y(2c6)=(c+5)

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||2c6|=|c+5|
x=+y , +x=y(2c6)=(c+5)
x=y , x=y(2c6)=(c+5)

2. Solve the two equations for c

7 additional steps

(2c-6)=(c+5)

Subtract from both sides:

(2c-6)-c=(c+5)-c

Group like terms:

(2c-c)-6=(c+5)-c

Simplify the arithmetic:

c-6=(c+5)-c

Group like terms:

c-6=(c-c)+5

Simplify the arithmetic:

c6=5

Add to both sides:

(c-6)+6=5+6

Simplify the arithmetic:

c=5+6

Simplify the arithmetic:

c=11

10 additional steps

(2c-6)=-(c+5)

Expand the parentheses:

(2c-6)=-c-5

Add to both sides:

(2c-6)+c=(-c-5)+c

Group like terms:

(2c+c)-6=(-c-5)+c

Simplify the arithmetic:

3c-6=(-c-5)+c

Group like terms:

3c-6=(-c+c)-5

Simplify the arithmetic:

3c6=5

Add to both sides:

(3c-6)+6=-5+6

Simplify the arithmetic:

3c=5+6

Simplify the arithmetic:

3c=1

Divide both sides by :

(3c)3=13

Simplify the fraction:

c=13

3. List the solutions

c=11,13
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
y=|2c6|
y=|c+5|
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.