Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

y = - \frac{9}{2}

y=-\frac{9}{2}

Mixed number form:

y = - 4 \frac{1}{2}

y=-4\frac{1}{2}

Decimal form:

y = - 4.5

y=-4.5

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
$| y + 5 | = | y + 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| y + 5 \| = \| y + 4 \|$
$x = + y$	$(y + 5) = (y + 4)$
$x = - y$	$(y + 5) = - (y + 4)$
$+ x = y$	$(y + 5) = (y + 4)$
$- x = y$	$- (y + 5) = (y + 4)$

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:

$\| x \| = \| y \|$	$\| y + 5 \| = \| y + 4 \|$
$x = + y$ , $+ x = y$	$(y + 5) = (y + 4)$
$x = - y$ , $- x = y$	$(y + 5) = - (y + 4)$

2. Solve the two equations for y

5 additional steps

$(y + 5) = (y + 4)$

Subtract from both sides:

$(y + 5) - y = (y + 4) - y$

Group like terms:

$(y - y) + 5 = (y + 4) - y$

Simplify the arithmetic:

$5 = (y + 4) - y$

Group like terms:

$5 = (y - y) + 4$

Simplify the arithmetic:

$5 = 4$

The statement is false:

$5 = 4$

The equation is false so it has no solution.

10 additional steps

$(y + 5) = - (y + 4)$

Expand the parentheses:

$(y + 5) = - y - 4$

Add to both sides:

$(y + 5) + y = (- y - 4) + y$

Group like terms:

$(y + y) + 5 = (- y - 4) + y$

Simplify the arithmetic:

$2 y + 5 = (- y - 4) + y$

Group like terms:

$2 y + 5 = (- y + y) - 4$

Simplify the arithmetic:

$2 y + 5 = - 4$

Subtract from both sides:

$(2 y + 5) - 5 = - 4 - 5$

Simplify the arithmetic:

$2 y = - 4 - 5$

Simplify the arithmetic:

$2 y = - 9$

Divide both sides by :

$\frac{(2 y)}{2} = \frac{- 9}{2}$

Simplify the fraction:

$y = \frac{- 9}{2}$

3. Graph

Each line represents the function of one side of the equation:
$y = | y + 5 |$
$y = | y + 4 |$
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. 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. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved