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Solution - Absolute value equations

Exact form:

y = \frac{7}{10}, - \frac{1}{2}

y=\frac{7}{10} , -\frac{1}{2}

Decimal form:

y = 0.7, - 0.5

y=0.7 , -0.5

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
$| - 2 y + 5 | = 2 | 4 y - 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 2 y + 5 \| = 2 \| 4 y - 1 \|$
$x = + y$	$(- 2 y + 5) = 2 (4 y - 1)$
$x = - y$	$(- 2 y + 5) = 2 (- (4 y - 1))$
$+ x = y$	$(- 2 y + 5) = 2 (4 y - 1)$
$- x = y$	$- (- 2 y + 5) = 2 (4 y - 1)$

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:

$\| x \| = \| y \|$	$\| - 2 y + 5 \| = 2 \| 4 y - 1 \|$
$x = + y$ , $+ x = y$	$(- 2 y + 5) = 2 (4 y - 1)$
$x = - y$ , $- x = y$	$(- 2 y + 5) = 2 (- (4 y - 1))$

2. Solve the two equations for y

14 additional steps

$(- 2 y + 5) = 2 \cdot (4 y - 1)$

Expand the parentheses:

$(- 2 y + 5) = 2 \cdot 4 y + 2 \cdot - 1$

Multiply the coefficients:

$(- 2 y + 5) = 8 y + 2 \cdot - 1$

Simplify the arithmetic:

$(- 2 y + 5) = 8 y - 2$

Subtract from both sides:

$(- 2 y + 5) - 8 y = (8 y - 2) - 8 y$

Group like terms:

$(- 2 y - 8 y) + 5 = (8 y - 2) - 8 y$

Simplify the arithmetic:

$- 10 y + 5 = (8 y - 2) - 8 y$

Group like terms:

$- 10 y + 5 = (8 y - 8 y) - 2$

Simplify the arithmetic:

$- 10 y + 5 = - 2$

Subtract from both sides:

$(- 10 y + 5) - 5 = - 2 - 5$

Simplify the arithmetic:

$- 10 y = - 2 - 5$

Simplify the arithmetic:

$- 10 y = - 7$

Divide both sides by :

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

Cancel out the negatives:

$\frac{10 y}{10} = \frac{- 7}{- 10}$

Simplify the fraction:

$y = \frac{- 7}{- 10}$

Cancel out the negatives:

$y = \frac{7}{10}$

15 additional steps

$(- 2 y + 5) = 2 \cdot (- (4 y - 1))$

Expand the parentheses:

$(- 2 y + 5) = 2 \cdot (- 4 y + 1)$

Expand the parentheses:

$(- 2 y + 5) = 2 \cdot - 4 y + 2 \cdot 1$

Multiply the coefficients:

$(- 2 y + 5) = - 8 y + 2 \cdot 1$

Simplify the arithmetic:

$(- 2 y + 5) = - 8 y + 2$

Add to both sides:

$(- 2 y + 5) + 8 y = (- 8 y + 2) + 8 y$

Group like terms:

$(- 2 y + 8 y) + 5 = (- 8 y + 2) + 8 y$

Simplify the arithmetic:

$6 y + 5 = (- 8 y + 2) + 8 y$

Group like terms:

$6 y + 5 = (- 8 y + 8 y) + 2$

Simplify the arithmetic:

$6 y + 5 = 2$

Subtract from both sides:

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

Simplify the arithmetic:

$6 y = 2 - 5$

Simplify the arithmetic:

$6 y = - 3$

Divide both sides by :

$\frac{(6 y)}{6} = \frac{- 3}{6}$

Simplify the fraction:

$y = \frac{- 3}{6}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

3. List the solutions

$y = \frac{7}{10}, - \frac{1}{2}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 2 y + 5 |$
$y = 2 | 4 y - 1 |$
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

Latest Related Drills Solved