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Solution - Absolute value equations

Exact form:

y = - 9, - \frac{1}{3}

y=-9 , -\frac{1}{3}

Decimal form:

y = - 9, - 0.333

y=-9 , -0.333

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
$| 2 y + 5 | = | y - 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 y + 5 \| = \| y - 4 \|$
$x = + y$	$(2 y + 5) = (y - 4)$
$x = - y$	$(2 y + 5) = - (y - 4)$
$+ x = y$	$(2 y + 5) = (y - 4)$
$- x = y$	$- (2 y + 5) = (y - 4)$

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:

$\| x \| = \| y \|$	$\| 2 y + 5 \| = \| y - 4 \|$
$x = + y$ , $+ x = y$	$(2 y + 5) = (y - 4)$
$x = - y$ , $- x = y$	$(2 y + 5) = - (y - 4)$

2. Solve the two equations for y

7 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$y + 5 = - 4$

Subtract from both sides:

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

Simplify the arithmetic:

$y = - 4 - 5$

Simplify the arithmetic:

$y = - 9$

10 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$3 y + 5 = (- y + 4) + y$

Group like terms:

$3 y + 5 = (- y + y) + 4$

Simplify the arithmetic:

$3 y + 5 = 4$

Subtract from both sides:

$(3 y + 5) - 5 = 4 - 5$

Simplify the arithmetic:

$3 y = 4 - 5$

Simplify the arithmetic:

$3 y = - 1$

Divide both sides by :

$\frac{(3 y)}{3} = \frac{- 1}{3}$

Simplify the fraction:

$y = \frac{- 1}{3}$

3. List the solutions

$y = - 9, - \frac{1}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 y + 5 |$
$y = | y - 4 |$
The equation is true where the two lines cross.

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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

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