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Solution - Absolute value equations

Exact form:

x = \frac{35}{19}, \frac{5}{29}

x=\frac{35}{19} , \frac{5}{29}

Mixed number form:

x = 1 \frac{16}{19}, \frac{5}{29}

x=1\frac{16}{19} , \frac{5}{29}

Decimal form:

x = 1.842, 0.172

x=1.842 , 0.172

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
$5 | x + 3 | = 4 | 6 x - 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$5 \| x + 3 \| = 4 \| 6 x - 5 \|$
$x = + y$	$5 (x + 3) = 4 (6 x - 5)$
$x = - y$	$5 (x + 3) = 4 (- (6 x - 5))$
$+ x = y$	$5 (x + 3) = 4 (6 x - 5)$
$- x = y$	$5 (- (x + 3)) = 4 (6 x - 5)$

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:

$\| x \| = \| y \|$	$5 \| x + 3 \| = 4 \| 6 x - 5 \|$
$x = + y$ , $+ x = y$	$5 (x + 3) = 4 (6 x - 5)$
$x = - y$ , $- x = y$	$5 (x + 3) = 4 (- (6 x - 5))$

2. Solve the two equations for x

16 additional steps

$5 \cdot (x + 3) = 4 \cdot (6 x - 5)$

Expand the parentheses:

$5 x + 5 \cdot 3 = 4 \cdot (6 x - 5)$

Simplify the arithmetic:

$5 x + 15 = 4 \cdot (6 x - 5)$

Expand the parentheses:

$5 x + 15 = 4 \cdot 6 x + 4 \cdot - 5$

Multiply the coefficients:

$5 x + 15 = 24 x + 4 \cdot - 5$

Simplify the arithmetic:

$5 x + 15 = 24 x - 20$

Subtract from both sides:

$(5 x + 15) - 24 x = (24 x - 20) - 24 x$

Group like terms:

$(5 x - 24 x) + 15 = (24 x - 20) - 24 x$

Simplify the arithmetic:

$- 19 x + 15 = (24 x - 20) - 24 x$

Group like terms:

$- 19 x + 15 = (24 x - 24 x) - 20$

Simplify the arithmetic:

$- 19 x + 15 = - 20$

Subtract from both sides:

$(- 19 x + 15) - 15 = - 20 - 15$

Simplify the arithmetic:

$- 19 x = - 20 - 15$

Simplify the arithmetic:

$- 19 x = - 35$

Divide both sides by :

$\frac{(- 19 x)}{- 19} = \frac{- 35}{- 19}$

Cancel out the negatives:

$\frac{19 x}{19} = \frac{- 35}{- 19}$

Simplify the fraction:

$x = \frac{- 35}{- 19}$

Cancel out the negatives:

$x = \frac{35}{19}$

15 additional steps

$5 \cdot (x + 3) = 4 \cdot (- (6 x - 5))$

Expand the parentheses:

$5 x + 5 \cdot 3 = 4 \cdot (- (6 x - 5))$

Simplify the arithmetic:

$5 x + 15 = 4 \cdot (- (6 x - 5))$

Expand the parentheses:

$5 x + 15 = 4 \cdot (- 6 x + 5)$

Expand the parentheses:

$5 x + 15 = 4 \cdot - 6 x + 4 \cdot 5$

Multiply the coefficients:

$5 x + 15 = - 24 x + 4 \cdot 5$

Simplify the arithmetic:

$5 x + 15 = - 24 x + 20$

Add to both sides:

$(5 x + 15) + 24 x = (- 24 x + 20) + 24 x$

Group like terms:

$(5 x + 24 x) + 15 = (- 24 x + 20) + 24 x$

Simplify the arithmetic:

$29 x + 15 = (- 24 x + 20) + 24 x$

Group like terms:

$29 x + 15 = (- 24 x + 24 x) + 20$

Simplify the arithmetic:

$29 x + 15 = 20$

Subtract from both sides:

$(29 x + 15) - 15 = 20 - 15$

Simplify the arithmetic:

$29 x = 20 - 15$

Simplify the arithmetic:

$29 x = 5$

Divide both sides by :

$\frac{(29 x)}{29} = \frac{5}{29}$

Simplify the fraction:

$x = \frac{5}{29}$

3. List the solutions

$x = \frac{35}{19}, \frac{5}{29}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 5 | x + 3 |$
$y = 4 | 6 x - 5 |$
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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