Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

y = - 3, - \frac{5}{3}

y=-3 , -\frac{5}{3}

Mixed number form:

y = - 3, - 1 \frac{2}{3}

y=-3 , -1\frac{2}{3}

Decimal form:

y = - 3, - 1.667

y=-3 , -1.667

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
$| 2 y + 4 | = | y + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 y + 4 \| = \| y + 1 \|$
$x = + y$	$(2 y + 4) = (y + 1)$
$x = - y$	$(2 y + 4) = - (y + 1)$
$+ x = y$	$(2 y + 4) = (y + 1)$
$- x = y$	$- (2 y + 4) = (y + 1)$

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:

$\| x \| = \| y \|$	$\| 2 y + 4 \| = \| y + 1 \|$
$x = + y$ , $+ x = y$	$(2 y + 4) = (y + 1)$
$x = - y$ , $- x = y$	$(2 y + 4) = - (y + 1)$

2. Solve the two equations for y

7 additional steps

$(2 y + 4) = (y + 1)$

Subtract from both sides:

$(2 y + 4) - y = (y + 1) - y$

Group like terms:

$(2 y - y) + 4 = (y + 1) - y$

Simplify the arithmetic:

$y + 4 = (y + 1) - y$

Group like terms:

$y + 4 = (y - y) + 1$

Simplify the arithmetic:

$y + 4 = 1$

Subtract from both sides:

$(y + 4) - 4 = 1 - 4$

Simplify the arithmetic:

$y = 1 - 4$

Simplify the arithmetic:

$y = - 3$

10 additional steps

$(2 y + 4) = - (y + 1)$

Expand the parentheses:

$(2 y + 4) = - y - 1$

Add to both sides:

$(2 y + 4) + y = (- y - 1) + y$

Group like terms:

$(2 y + y) + 4 = (- y - 1) + y$

Simplify the arithmetic:

$3 y + 4 = (- y - 1) + y$

Group like terms:

$3 y + 4 = (- y + y) - 1$

Simplify the arithmetic:

$3 y + 4 = - 1$

Subtract from both sides:

$(3 y + 4) - 4 = - 1 - 4$

Simplify the arithmetic:

$3 y = - 1 - 4$

Simplify the arithmetic:

$3 y = - 5$

Divide both sides by :

$\frac{(3 y)}{3} = \frac{- 5}{3}$

Simplify the fraction:

$y = \frac{- 5}{3}$

3. List the solutions

$y = - 3, - \frac{5}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 y + 4 |$
$y = | y + 1 |$
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