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Solution - Absolute value equations

Exact form:

y = - 4, - \frac{1}{3}

y=-4 , -\frac{1}{3}

Decimal form:

y = - 4, - 0.333

y=-4 , -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
$| 5 y - 2 | = | 7 y + 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 y - 2 \| = \| 7 y + 6 \|$
$x = + y$	$(5 y - 2) = (7 y + 6)$
$x = - y$	$(5 y - 2) = - (7 y + 6)$
$+ x = y$	$(5 y - 2) = (7 y + 6)$
$- x = y$	$- (5 y - 2) = (7 y + 6)$

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:

$\| x \| = \| y \|$	$\| 5 y - 2 \| = \| 7 y + 6 \|$
$x = + y$ , $+ x = y$	$(5 y - 2) = (7 y + 6)$
$x = - y$ , $- x = y$	$(5 y - 2) = - (7 y + 6)$

2. Solve the two equations for y

13 additional steps

$(5 y - 2) = (7 y + 6)$

Subtract from both sides:

$(5 y - 2) - 7 y = (7 y + 6) - 7 y$

Group like terms:

$(5 y - 7 y) - 2 = (7 y + 6) - 7 y$

Simplify the arithmetic:

$- 2 y - 2 = (7 y + 6) - 7 y$

Group like terms:

$- 2 y - 2 = (7 y - 7 y) + 6$

Simplify the arithmetic:

$- 2 y - 2 = 6$

Add to both sides:

$(- 2 y - 2) + 2 = 6 + 2$

Simplify the arithmetic:

$- 2 y = 6 + 2$

Simplify the arithmetic:

$- 2 y = 8$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Move the negative sign from the denominator to the numerator:

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

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

$(5 y - 2) = - (7 y + 6)$

Expand the parentheses:

$(5 y - 2) = - 7 y - 6$

Add to both sides:

$(5 y - 2) + 7 y = (- 7 y - 6) + 7 y$

Group like terms:

$(5 y + 7 y) - 2 = (- 7 y - 6) + 7 y$

Simplify the arithmetic:

$12 y - 2 = (- 7 y - 6) + 7 y$

Group like terms:

$12 y - 2 = (- 7 y + 7 y) - 6$

Simplify the arithmetic:

$12 y - 2 = - 6$

Add to both sides:

$(12 y - 2) + 2 = - 6 + 2$

Simplify the arithmetic:

$12 y = - 6 + 2$

Simplify the arithmetic:

$12 y = - 4$

Divide both sides by :

$\frac{(12 y)}{12} = \frac{- 4}{12}$

Simplify the fraction:

$y = \frac{- 4}{12}$

Find the greatest common factor of the numerator and denominator:

$y = \frac{(- 1 \cdot 4)}{(3 \cdot 4)}$

Factor out and cancel the greatest common factor:

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

3. List the solutions

$y = - 4, - \frac{1}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 y - 2 |$
$y = | 7 y + 6 |$
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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