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Solution - Absolute value equations

Exact form:

y = \frac{5}{2}, - \frac{11}{10}

y=\frac{5}{2} , -\frac{11}{10}

Mixed number form:

y = 2 \frac{1}{2}, - 1 \frac{1}{10}

y=2\frac{1}{2} , -1\frac{1}{10}

Decimal form:

y = 2.5, - 1.1

y=2.5 , -1.1

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
$| 6 y + 3 | = 4 | y + 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 6 y + 3 \| = 4 \| y + 2 \|$
$x = + y$	$(6 y + 3) = 4 (y + 2)$
$x = - y$	$(6 y + 3) = 4 (- (y + 2))$
$+ x = y$	$(6 y + 3) = 4 (y + 2)$
$- x = y$	$- (6 y + 3) = 4 (y + 2)$

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:

$\| x \| = \| y \|$	$\| 6 y + 3 \| = 4 \| y + 2 \|$
$x = + y$ , $+ x = y$	$(6 y + 3) = 4 (y + 2)$
$x = - y$ , $- x = y$	$(6 y + 3) = 4 (- (y + 2))$

2. Solve the two equations for y

11 additional steps

$(6 y + 3) = 4 \cdot (y + 2)$

Expand the parentheses:

$(6 y + 3) = 4 y + 4 \cdot 2$

Simplify the arithmetic:

$(6 y + 3) = 4 y + 8$

Subtract from both sides:

$(6 y + 3) - 4 y = (4 y + 8) - 4 y$

Group like terms:

$(6 y - 4 y) + 3 = (4 y + 8) - 4 y$

Simplify the arithmetic:

$2 y + 3 = (4 y + 8) - 4 y$

Group like terms:

$2 y + 3 = (4 y - 4 y) + 8$

Simplify the arithmetic:

$2 y + 3 = 8$

Subtract from both sides:

$(2 y + 3) - 3 = 8 - 3$

Simplify the arithmetic:

$2 y = 8 - 3$

Simplify the arithmetic:

$2 y = 5$

Divide both sides by :

$\frac{(2 y)}{2} = \frac{5}{2}$

Simplify the fraction:

$y = \frac{5}{2}$

14 additional steps

$(6 y + 3) = 4 \cdot (- (y + 2))$

Expand the parentheses:

$(6 y + 3) = 4 \cdot (- y - 2)$

$(6 y + 3) = 4 \cdot - y + 4 \cdot - 2$

Group like terms:

$(6 y + 3) = (4 \cdot - 1) y + 4 \cdot - 2$

Multiply the coefficients:

$(6 y + 3) = - 4 y + 4 \cdot - 2$

Simplify the arithmetic:

$(6 y + 3) = - 4 y - 8$

Add to both sides:

$(6 y + 3) + 4 y = (- 4 y - 8) + 4 y$

Group like terms:

$(6 y + 4 y) + 3 = (- 4 y - 8) + 4 y$

Simplify the arithmetic:

$10 y + 3 = (- 4 y - 8) + 4 y$

Group like terms:

$10 y + 3 = (- 4 y + 4 y) - 8$

Simplify the arithmetic:

$10 y + 3 = - 8$

Subtract from both sides:

$(10 y + 3) - 3 = - 8 - 3$

Simplify the arithmetic:

$10 y = - 8 - 3$

Simplify the arithmetic:

$10 y = - 11$

Divide both sides by :

$\frac{(10 y)}{10} = \frac{- 11}{10}$

Simplify the fraction:

$y = \frac{- 11}{10}$

3. List the solutions

$y = \frac{5}{2}, - \frac{11}{10}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 6 y + 3 |$
$y = 4 | y + 2 |$
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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