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Solution - Absolute value equations

Exact form:

y = - 0.152, 0.106

y=-0.152 , 0.106

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
$| 0.07 y - 0.05 | = | 0.4 y |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 0.07 y - 0.05 \| = \| 0.4 y \|$
$x = + y$	$(0.07 y - 0.05) = (0.4 y)$
$x = - y$	$(0.07 y - 0.05) = - (0.4 y)$
$+ x = y$	$(0.07 y - 0.05) = (0.4 y)$
$- x = y$	$- (0.07 y - 0.05) = (0.4 y)$

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:

$\| x \| = \| y \|$	$\| 0.07 y - 0.05 \| = \| 0.4 y \|$
$x = + y$ , $+ x = y$	$(0.07 y - 0.05) = (0.4 y)$
$x = - y$ , $- x = y$	$(0.07 y - 0.05) = - (0.4 y)$

2. Solve the two equations for y

11 additional steps

$(0.07 y - 0.05) = 0.4 y$

Subtract from both sides:

$(0.07 y - 0.05) - 0.4 y = (0.4 y) - 0.4 y$

Group like terms:

$(0.07 y - 0.4 y) - 0.05 = (0.4 y) - 0.4 y$

Simplify the arithmetic:

$- 0.33 y - 0.05 = (0.4 y) - 0.4 y$

Simplify the arithmetic:

$- 0.33 y - 0.05 = 0$

Add to both sides:

$(- 0.33 y - 0.05) + 0.05 = 0 + 0.05$

Simplify the arithmetic:

$- 0.33 y = 0 + 0.05$

Simplify the arithmetic:

$- 0.33 y = 0.05$

Divide both sides by :

$\frac{(- 0.33 y)}{- 0.33} = \frac{0.05}{- 0.33}$

Cancel out the negatives:

$\frac{0.33 y}{0.33} = \frac{0.05}{- 0.33}$

Simplify the arithmetic:

$y = \frac{0.05}{- 0.33}$

Move the negative sign from the denominator to the numerator:

$y = \frac{- 0.05}{0.33}$

Simplify the arithmetic:

$y = - 0.1515$

8 additional steps

$(0.07 y - 0.05) = - 0.4 y$

Add to both sides:

$(0.07 y - 0.05) + 0.05 = (- 0.4 y) + 0.05$

Simplify the arithmetic:

$0.07 y = (- 0.4 y) + 0.05$

Add to both sides:

$(0.07 y) + 0.4 y = ((- 0.4 y) + 0.05) + 0.4 y$

Simplify the arithmetic:

$0.47 y = ((- 0.4 y) + 0.05) + 0.4 y$

Group like terms:

$0.47 y = (- 0.4 y + 0.4 y) + 0.05$

Simplify the arithmetic:

$0.47 y = 0.05$

Divide both sides by :

$\frac{(0.47 y)}{0.47} = \frac{0.05}{0.47}$

Simplify the arithmetic:

$y = \frac{0.05}{0.47}$

Simplify the arithmetic:

$y = 0.1064$

3. List the solutions

$y = - 0.152, 0.106$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 0.07 y - 0.05 |$
$y = | 0.4 y |$
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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