Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

y = \frac{7}{2}

y=\frac{7}{2}

Mixed number form:

y = 3 \frac{1}{2}

y=3\frac{1}{2}

Decimal form:

y = 3.5

y=3.5

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
$| y | = | - y + 7 |$
without the absolute value bars:

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

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:

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

2. Solve the two equations for y

5 additional steps

$y = (- y + 7)$

Add to both sides:

$y + y = (- y + 7) + y$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$2 y = 7$

Divide both sides by :

$\frac{(2 y)}{2} = \frac{7}{2}$

Simplify the fraction:

$y = \frac{7}{2}$

5 additional steps

$y = - (- y + 7)$

Expand the parentheses:

$y = y - 7$

Subtract from both sides:

$y - y = (y - 7) - y$

Simplify the arithmetic:

$0 = (y - 7) - y$

Group like terms:

$0 = (y - y) - 7$

Simplify the arithmetic:

$0 = - 7$

The statement is false:

$0 = - 7$

The equation is false so it has no solution.

3. List the solutions

$y = \frac{7}{2}$
(1 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | y |$
$y = | - y + 7 |$
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