Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

y = - 13, - 1

y=-13 , -1

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
$| 2 y + 5 | = \frac{1}{2} | 3 y - 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 y + 5 \| = \frac{1}{2} \| 3 y - 3 \|$
$x = + y$	$(2 y + 5) = \frac{1}{2} (3 y - 3)$
$x = - y$	$(2 y + 5) = \frac{1}{2} (- (3 y - 3))$
$+ x = y$	$(2 y + 5) = \frac{1}{2} (3 y - 3)$
$- x = y$	$- (2 y + 5) = \frac{1}{2} (3 y - 3)$

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:

$\| x \| = \| y \|$	$\| 2 y + 5 \| = \frac{1}{2} \| 3 y - 3 \|$
$x = + y$ , $+ x = y$	$(2 y + 5) = \frac{1}{2} (3 y - 3)$
$x = - y$ , $- x = y$	$(2 y + 5) = \frac{1}{2} (- (3 y - 3))$

2. Solve the two equations for y

23 additional steps

$(2 y + 5) = \frac{1}{2} \cdot (3 y - 3)$

Multiply the fraction(s):

$(2 y + 5) = \frac{(1 \cdot (3 y - 3))}{2}$

Break up the fraction:

$(2 y + 5) = \frac{3 y}{2} + \frac{- 3}{2}$

Subtract from both sides:

$(2 y + 5) - \frac{3 y}{2} = (\frac{3 y}{2} + \frac{- 3}{2}) - \frac{3 y}{2}$

Group like terms:

$(2 y + \frac{- 3}{2} y) + 5 = (\frac{3 y}{2} + \frac{- 3}{2}) - \frac{3 y}{2}$

Group the coefficients:

$(2 + \frac{- 3}{2}) y + 5 = (\frac{3 y}{2} + \frac{- 3}{2}) - \frac{3 y}{2}$

Convert the integer into a fraction:

$(\frac{4}{2} + \frac{- 3}{2}) y + 5 = (\frac{3 y}{2} + \frac{- 3}{2}) - \frac{3 y}{2}$

Combine the fractions:

$\frac{(4 - 3)}{2} y + 5 = (\frac{3 y}{2} + \frac{- 3}{2}) - \frac{3 y}{2}$

Combine the numerators:

$\frac{1}{2} y + 5 = (\frac{3 y}{2} + \frac{- 3}{2}) - \frac{3 y}{2}$

Group like terms:

$\frac{1}{2} \cdot y + 5 = (\frac{3 y}{2} + \frac{- 3}{2} y) + \frac{- 3}{2}$

Combine the fractions:

$\frac{1}{2} \cdot y + 5 = \frac{(3 - 3)}{2} y + \frac{- 3}{2}$

Combine the numerators:

$\frac{1}{2} \cdot y + 5 = \frac{0}{2} y + \frac{- 3}{2}$

Reduce the zero numerator:

$\frac{1}{2} y + 5 = 0 y + \frac{- 3}{2}$

Simplify the arithmetic:

$\frac{1}{2} y + 5 = \frac{- 3}{2}$

Subtract from both sides:

$(\frac{1}{2} y + 5) - 5 = (\frac{- 3}{2}) - 5$

Simplify the arithmetic:

$\frac{1}{2} y = (\frac{- 3}{2}) - 5$

Convert the integer into a fraction:

$\frac{1}{2} y = \frac{- 3}{2} + \frac{- 10}{2}$

Combine the fractions:

$\frac{1}{2} y = \frac{(- 3 - 10)}{2}$

Combine the numerators:

$\frac{1}{2} y = \frac{- 13}{2}$

Multiply both sides by inverse fraction :

$(\frac{1}{2} y) \cdot \frac{2}{1} = (\frac{- 13}{2}) \cdot \frac{2}{1}$

Group like terms:

$(\frac{1}{2} \cdot 2) y = (\frac{- 13}{2}) \cdot \frac{2}{1}$

Multiply the coefficients:

$\frac{(1 \cdot 2)}{2} y = (\frac{- 13}{2}) \cdot \frac{2}{1}$

Simplify the fraction:

$y = (\frac{- 13}{2}) \cdot \frac{2}{1}$

Multiply the fraction(s):

$y = \frac{(- 13 \cdot 2)}{2}$

Simplify the arithmetic:

$y = - 13$

24 additional steps

$(2 y + 5) = \frac{1}{2} \cdot (- (3 y - 3))$

Multiply the fraction(s):

$(2 y + 5) = \frac{(1 \cdot (- (3 y - 3)))}{2}$

Expand the parentheses:

$(2 y + 5) = \frac{(- 3 y + 3)}{2}$

Break up the fraction:

$(2 y + 5) = \frac{- 3 y}{2} + \frac{3}{2}$

Add to both sides:

$(2 y + 5) + \frac{3}{2} \cdot y = (\frac{- 3 y}{2} + \frac{3}{2}) + \frac{3}{2} y$

Group like terms:

$(2 y + \frac{3}{2} \cdot y) + 5 = (\frac{- 3 y}{2} + \frac{3}{2}) + \frac{3}{2} y$

Group the coefficients:

$(2 + \frac{3}{2}) y + 5 = (\frac{- 3 y}{2} + \frac{3}{2}) + \frac{3}{2} y$

Convert the integer into a fraction:

$(\frac{4}{2} + \frac{3}{2}) y + 5 = (\frac{- 3 y}{2} + \frac{3}{2}) + \frac{3}{2} y$

Combine the fractions:

$\frac{(4 + 3)}{2} \cdot y + 5 = (\frac{- 3 y}{2} + \frac{3}{2}) + \frac{3}{2} y$

Combine the numerators:

$\frac{7}{2} \cdot y + 5 = (\frac{- 3 y}{2} + \frac{3}{2}) + \frac{3}{2} y$

Group like terms:

$\frac{7}{2} \cdot y + 5 = (\frac{- 3 y}{2} + \frac{3}{2} y) + \frac{3}{2}$

Combine the fractions:

$\frac{7}{2} \cdot y + 5 = \frac{(- 3 + 3)}{2} y + \frac{3}{2}$

Combine the numerators:

$\frac{7}{2} \cdot y + 5 = \frac{0}{2} y + \frac{3}{2}$

Reduce the zero numerator:

$\frac{7}{2} y + 5 = 0 y + \frac{3}{2}$

Simplify the arithmetic:

$\frac{7}{2} y + 5 = \frac{3}{2}$

Subtract from both sides:

$(\frac{7}{2} y + 5) - 5 = (\frac{3}{2}) - 5$

Simplify the arithmetic:

$\frac{7}{2} y = (\frac{3}{2}) - 5$

Convert the integer into a fraction:

$\frac{7}{2} y = \frac{3}{2} + \frac{- 10}{2}$

Combine the fractions:

$\frac{7}{2} y = \frac{(3 - 10)}{2}$

Combine the numerators:

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

Multiply both sides by inverse fraction :

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

Group like terms:

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

Multiply the coefficients:

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

Simplify the fraction:

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

Multiply the fraction(s):

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

Simplify the arithmetic:

$y = - 1$

3. List the solutions

$y = - 13, - 1$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 y + 5 |$
$y = \frac{1}{2} | 3 y - 3 |$
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