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Solution - Absolute value equations

Exact form:

y = 4, - \frac{6}{7}

y=4 , -\frac{6}{7}

Decimal form:

y = 4, - 0.857

y=4 , -0.857

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
$| 3 y + 5 | = | 4 y + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 y + 5 \| = \| 4 y + 1 \|$
$x = + y$	$(3 y + 5) = (4 y + 1)$
$x = - y$	$(3 y + 5) = - (4 y + 1)$
$+ x = y$	$(3 y + 5) = (4 y + 1)$
$- x = y$	$- (3 y + 5) = (4 y + 1)$

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:

$\| x \| = \| y \|$	$\| 3 y + 5 \| = \| 4 y + 1 \|$
$x = + y$ , $+ x = y$	$(3 y + 5) = (4 y + 1)$
$x = - y$ , $- x = y$	$(3 y + 5) = - (4 y + 1)$

2. Solve the two equations for y

10 additional steps

$(3 y + 5) = (4 y + 1)$

Subtract from both sides:

$(3 y + 5) - 4 y = (4 y + 1) - 4 y$

Group like terms:

$(3 y - 4 y) + 5 = (4 y + 1) - 4 y$

Simplify the arithmetic:

$- y + 5 = (4 y + 1) - 4 y$

Group like terms:

$- y + 5 = (4 y - 4 y) + 1$

Simplify the arithmetic:

$- y + 5 = 1$

Subtract from both sides:

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

Simplify the arithmetic:

$- y = 1 - 5$

Simplify the arithmetic:

$- y = - 4$

Multiply both sides by :

$- y \cdot - 1 = - 4 \cdot - 1$

Remove the one(s):

$y = - 4 \cdot - 1$

Simplify the arithmetic:

$y = 4$

10 additional steps

$(3 y + 5) = - (4 y + 1)$

Expand the parentheses:

$(3 y + 5) = - 4 y - 1$

Add to both sides:

$(3 y + 5) + 4 y = (- 4 y - 1) + 4 y$

Group like terms:

$(3 y + 4 y) + 5 = (- 4 y - 1) + 4 y$

Simplify the arithmetic:

$7 y + 5 = (- 4 y - 1) + 4 y$

Group like terms:

$7 y + 5 = (- 4 y + 4 y) - 1$

Simplify the arithmetic:

$7 y + 5 = - 1$

Subtract from both sides:

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

Simplify the arithmetic:

$7 y = - 1 - 5$

Simplify the arithmetic:

$7 y = - 6$

Divide both sides by :

$\frac{(7 y)}{7} = \frac{- 6}{7}$

Simplify the fraction:

$y = \frac{- 6}{7}$

3. List the solutions

$y = 4, - \frac{6}{7}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 y + 5 |$
$y = | 4 y + 1 |$
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

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