Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

y = 4

y=4

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
$| 2 y - 6 | = | - 2 y + 10 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 y - 6 \| = \| - 2 y + 10 \|$
$x = + y$	$(2 y - 6) = (- 2 y + 10)$
$x = - y$	$(2 y - 6) = - (- 2 y + 10)$
$+ x = y$	$(2 y - 6) = (- 2 y + 10)$
$- x = y$	$- (2 y - 6) = (- 2 y + 10)$

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:

$\| x \| = \| y \|$	$\| 2 y - 6 \| = \| - 2 y + 10 \|$
$x = + y$ , $+ x = y$	$(2 y - 6) = (- 2 y + 10)$
$x = - y$ , $- x = y$	$(2 y - 6) = - (- 2 y + 10)$

2. Solve the two equations for y

11 additional steps

$(2 y - 6) = (- 2 y + 10)$

Add to both sides:

$(2 y - 6) + 2 y = (- 2 y + 10) + 2 y$

Group like terms:

$(2 y + 2 y) - 6 = (- 2 y + 10) + 2 y$

Simplify the arithmetic:

$4 y - 6 = (- 2 y + 10) + 2 y$

Group like terms:

$4 y - 6 = (- 2 y + 2 y) + 10$

Simplify the arithmetic:

$4 y - 6 = 10$

Add to both sides:

$(4 y - 6) + 6 = 10 + 6$

Simplify the arithmetic:

$4 y = 10 + 6$

Simplify the arithmetic:

$4 y = 16$

Divide both sides by :

$\frac{(4 y)}{4} = \frac{16}{4}$

Simplify the fraction:

$y = \frac{16}{4}$

Find the greatest common factor of the numerator and denominator:

$y = \frac{(4 \cdot 4)}{(1 \cdot 4)}$

Factor out and cancel the greatest common factor:

$y = 4$

6 additional steps

$(2 y - 6) = - (- 2 y + 10)$

Expand the parentheses:

$(2 y - 6) = 2 y - 10$

Subtract from both sides:

$(2 y - 6) - 2 y = (2 y - 10) - 2 y$

Group like terms:

$(2 y - 2 y) - 6 = (2 y - 10) - 2 y$

Simplify the arithmetic:

$- 6 = (2 y - 10) - 2 y$

Group like terms:

$- 6 = (2 y - 2 y) - 10$

Simplify the arithmetic:

$- 6 = - 10$

The statement is false:

$- 6 = - 10$

The equation is false so it has no solution.

3. List the solutions

$y = 4$
(1 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 y - 6 |$
$y = | - 2 y + 10 |$
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 y

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 y

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved