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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
$| 2 y + 5 | = 0.5 | 3 y - 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 y + 5 \| = 0.5 \| 3 y - 3 \|$
$x = + y$	$(2 y + 5) = 0.5 (3 y - 3)$
$x = - y$	$(2 y + 5) = 0.5 (- (3 y - 3))$
$+ x = y$	$(2 y + 5) = 0.5 (3 y - 3)$
$- x = y$	$- (2 y + 5) = 0.5 (3 y - 3)$

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:

$\| x \| = \| y \|$	$\| 2 y + 5 \| = 0.5 \| 3 y - 3 \|$
$x = + y$ , $+ x = y$	$(2 y + 5) = 0.5 (3 y - 3)$
$x = - y$ , $- x = y$	$(2 y + 5) = 0.5 (- (3 y - 3))$

2. Solve the two equations for y

13 additional steps

$(2 y + 5) = 0.5 \cdot (3 y - 3)$

Expand the parentheses:

$(2 y + 5) = 0.5 \cdot 3 y + 0.5 \cdot - 3$

Multiply the coefficients:

$(2 y + 5) = 1.5 y + 0.5 \cdot - 3$

Simplify the arithmetic:

$(2 y + 5) = 1.5 y - 1.5$

Subtract from both sides:

$(2 y + 5) - 1.5 y = (1.5 y - 1.5) - 1.5 y$

Group like terms:

$(2 y - 1.5 y) + 5 = (1.5 y - 1.5) - 1.5 y$

Simplify the arithmetic:

$0.5 y + 5 = (1.5 y - 1.5) - 1.5 y$

Group like terms:

$0.5 y + 5 = (1.5 y - 1.5 y) - 1.5$

Simplify the arithmetic:

$0.5 y + 5 = - 1.5$

Subtract from both sides:

$(0.5 y + 5) - 5 = - 1.5 - 5$

Simplify the arithmetic:

$0.5 y = - 1.5 - 5$

Simplify the arithmetic:

$0.5 y = - 6.5$

Divide both sides by :

$\frac{(0.5 y)}{0.5} = \frac{- 6.5}{0.5}$

Simplify the arithmetic:

$y = \frac{- 6.5}{0.5}$

Simplify the arithmetic:

$y = - 13$

14 additional steps

$(2 y + 5) = 0.5 \cdot (- (3 y - 3))$

Expand the parentheses:

$(2 y + 5) = 0.5 \cdot (- 3 y + 3)$

Expand the parentheses:

$(2 y + 5) = 0.5 \cdot - 3 y + 0.5 \cdot 3$

Multiply the coefficients:

$(2 y + 5) = - 1.5 y + 0.5 \cdot 3$

Simplify the arithmetic:

$(2 y + 5) = - 1.5 y + 1.5$

Add to both sides:

$(2 y + 5) + 1.5 y = (- 1.5 y + 1.5) + 1.5 y$

Group like terms:

$(2 y + 1.5 y) + 5 = (- 1.5 y + 1.5) + 1.5 y$

Simplify the arithmetic:

$3.5 y + 5 = (- 1.5 y + 1.5) + 1.5 y$

Group like terms:

$3.5 y + 5 = (- 1.5 y + 1.5 y) + 1.5$

Simplify the arithmetic:

$3.5 y + 5 = 1.5$

Subtract from both sides:

$(3.5 y + 5) - 5 = 1.5 - 5$

Simplify the arithmetic:

$3.5 y = 1.5 - 5$

Simplify the arithmetic:

$3.5 y = - 3.5$

Divide both sides by :

$\frac{(3.5 y)}{3.5} = \frac{- 3.5}{3.5}$

Simplify the arithmetic:

$y = \frac{- 3.5}{3.5}$

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:
$y = | 2 y + 5 |$
$y = 0.5 | 3 y - 3 |$
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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