Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

c = - 5, 3

c=-5 , 3

See steps

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
$| x | = | y |$ → $x = \pm y$ and $| x | = | y |$ → $\pm x = y$
to write all four options of the equation
$| c - 7 | = | 2 c - 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| c - 7 \| = \| 2 c - 2 \|$
$x = + y$	$(c - 7) = (2 c - 2)$
$x = - y$	$(c - 7) = - (2 c - 2)$
$+ x = y$	$(c - 7) = (2 c - 2)$
$- x = y$	$- (c - 7) = (2 c - 2)$

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:

$\| x \| = \| y \|$	$\| c - 7 \| = \| 2 c - 2 \|$
$x = + y$ , $+ x = y$	$(c - 7) = (2 c - 2)$
$x = - y$ , $- x = y$	$(c - 7) = - (2 c - 2)$

2. Solve the two equations for c

10 additional steps

$(c - 7) = (2 c - 2)$

Subtract from both sides:

$(c - 7) - 2 c = (2 c - 2) - 2 c$

Group like terms:

$(c - 2 c) - 7 = (2 c - 2) - 2 c$

Simplify the arithmetic:

$- c - 7 = (2 c - 2) - 2 c$

Group like terms:

$- c - 7 = (2 c - 2 c) - 2$

Simplify the arithmetic:

$- c - 7 = - 2$

Add to both sides:

$(- c - 7) + 7 = - 2 + 7$

Simplify the arithmetic:

$- c = - 2 + 7$

Simplify the arithmetic:

$- c = 5$

Multiply both sides by :

$- c \cdot - 1 = 5 \cdot - 1$

Remove the one(s):

$c = 5 \cdot - 1$

Simplify the arithmetic:

$c = - 5$

12 additional steps

$(c - 7) = - (2 c - 2)$

Expand the parentheses:

$(c - 7) = - 2 c + 2$

Add to both sides:

$(c - 7) + 2 c = (- 2 c + 2) + 2 c$

Group like terms:

$(c + 2 c) - 7 = (- 2 c + 2) + 2 c$

Simplify the arithmetic:

$3 c - 7 = (- 2 c + 2) + 2 c$

Group like terms:

$3 c - 7 = (- 2 c + 2 c) + 2$

Simplify the arithmetic:

$3 c - 7 = 2$

Add to both sides:

$(3 c - 7) + 7 = 2 + 7$

Simplify the arithmetic:

$3 c = 2 + 7$

Simplify the arithmetic:

$3 c = 9$

Divide both sides by :

$\frac{(3 c)}{3} = \frac{9}{3}$

Simplify the fraction:

$c = \frac{9}{3}$

Find the greatest common factor of the numerator and denominator:

$c = \frac{(3 \cdot 3)}{(1 \cdot 3)}$

Factor out and cancel the greatest common factor:

$c = 3$

3. List the solutions

$c = - 5, 3$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | c - 7 |$
$y = | 2 c - 2 |$
The equation is true where the two lines cross.

How did we do?

Please leave us feedback.

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.

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for c

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for c

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved