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Solution - Absolute value equations

Exact form:

x = \frac{15}{7}, 15

x=\frac{15}{7} , 15

Mixed number form:

x = 2 \frac{1}{7}, 15

x=2\frac{1}{7} , 15

Decimal form:

x = 2.143, 15

x=2.143 , 15

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
$| - 4 x + 15 | = | 3 x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 4 x + 15 \| = \| 3 x \|$
$x = + y$	$(- 4 x + 15) = (3 x)$
$x = - y$	$(- 4 x + 15) = - (3 x)$
$+ x = y$	$(- 4 x + 15) = (3 x)$
$- x = y$	$- (- 4 x + 15) = (3 x)$

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:

$\| x \| = \| y \|$	$\| - 4 x + 15 \| = \| 3 x \|$
$x = + y$ , $+ x = y$	$(- 4 x + 15) = (3 x)$
$x = - y$ , $- x = y$	$(- 4 x + 15) = - (3 x)$

2. Solve the two equations for x

10 additional steps

$(- 4 x + 15) = 3 x$

Subtract from both sides:

$(- 4 x + 15) - 3 x = (3 x) - 3 x$

Group like terms:

$(- 4 x - 3 x) + 15 = (3 x) - 3 x$

Simplify the arithmetic:

$- 7 x + 15 = (3 x) - 3 x$

Simplify the arithmetic:

$- 7 x + 15 = 0$

Subtract from both sides:

$(- 7 x + 15) - 15 = 0 - 15$

Simplify the arithmetic:

$- 7 x = 0 - 15$

Simplify the arithmetic:

$- 7 x = - 15$

Divide both sides by :

$\frac{(- 7 x)}{- 7} = \frac{- 15}{- 7}$

Cancel out the negatives:

$\frac{7 x}{7} = \frac{- 15}{- 7}$

Simplify the fraction:

$x = \frac{- 15}{- 7}$

Cancel out the negatives:

$x = \frac{15}{7}$

8 additional steps

$(- 4 x + 15) = - 3 x$

Subtract from both sides:

$(- 4 x + 15) - 15 = (- 3 x) - 15$

Simplify the arithmetic:

$- 4 x = (- 3 x) - 15$

Add to both sides:

$(- 4 x) + 3 x = ((- 3 x) - 15) + 3 x$

Simplify the arithmetic:

$- x = ((- 3 x) - 15) + 3 x$

Group like terms:

$- x = (- 3 x + 3 x) - 15$

Simplify the arithmetic:

$- x = - 15$

Multiply both sides by :

$- x \cdot - 1 = - 15 \cdot - 1$

Remove the one(s):

$x = - 15 \cdot - 1$

Simplify the arithmetic:

$x = 15$

3. List the solutions

$x = \frac{15}{7}, 15$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 4 x + 15 |$
$y = | 3 x |$
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 x

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 x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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