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Solution - Absolute value equations

Exact form:

y = - 4, \frac{3}{4}

y=-4 , \frac{3}{4}

Decimal form:

y = - 4, 0.75

y=-4 , 0.75

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
$| 5 y + 1 | = | 3 y - 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 y + 1 \| = \| 3 y - 7 \|$
$x = + y$	$(5 y + 1) = (3 y - 7)$
$x = - y$	$(5 y + 1) = - (3 y - 7)$
$+ x = y$	$(5 y + 1) = (3 y - 7)$
$- x = y$	$- (5 y + 1) = (3 y - 7)$

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:

$\| x \| = \| y \|$	$\| 5 y + 1 \| = \| 3 y - 7 \|$
$x = + y$ , $+ x = y$	$(5 y + 1) = (3 y - 7)$
$x = - y$ , $- x = y$	$(5 y + 1) = - (3 y - 7)$

2. Solve the two equations for y

11 additional steps

$(5 y + 1) = (3 y - 7)$

Subtract from both sides:

$(5 y + 1) - 3 y = (3 y - 7) - 3 y$

Group like terms:

$(5 y - 3 y) + 1 = (3 y - 7) - 3 y$

Simplify the arithmetic:

$2 y + 1 = (3 y - 7) - 3 y$

Group like terms:

$2 y + 1 = (3 y - 3 y) - 7$

Simplify the arithmetic:

$2 y + 1 = - 7$

Subtract from both sides:

$(2 y + 1) - 1 = - 7 - 1$

Simplify the arithmetic:

$2 y = - 7 - 1$

Simplify the arithmetic:

$2 y = - 8$

Divide both sides by :

$\frac{(2 y)}{2} = \frac{- 8}{2}$

Simplify the fraction:

$y = \frac{- 8}{2}$

Find the greatest common factor of the numerator and denominator:

$y = \frac{(- 4 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$y = - 4$

12 additional steps

$(5 y + 1) = - (3 y - 7)$

Expand the parentheses:

$(5 y + 1) = - 3 y + 7$

Add to both sides:

$(5 y + 1) + 3 y = (- 3 y + 7) + 3 y$

Group like terms:

$(5 y + 3 y) + 1 = (- 3 y + 7) + 3 y$

Simplify the arithmetic:

$8 y + 1 = (- 3 y + 7) + 3 y$

Group like terms:

$8 y + 1 = (- 3 y + 3 y) + 7$

Simplify the arithmetic:

$8 y + 1 = 7$

Subtract from both sides:

$(8 y + 1) - 1 = 7 - 1$

Simplify the arithmetic:

$8 y = 7 - 1$

Simplify the arithmetic:

$8 y = 6$

Divide both sides by :

$\frac{(8 y)}{8} = \frac{6}{8}$

Simplify the fraction:

$y = \frac{6}{8}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$y = \frac{3}{4}$

3. List the solutions

$y = - 4, \frac{3}{4}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 y + 1 |$
$y = | 3 y - 7 |$
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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