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Solution - Absolute value equations

Exact form:

x = - \frac{11}{7}, \frac{7}{13}

x=-\frac{11}{7} , \frac{7}{13}

Mixed number form:

x = - 1 \frac{4}{7}, \frac{7}{13}

x=-1\frac{4}{7} , \frac{7}{13}

Decimal form:

x = - 1.571, 0.538

x=-1.571 , 0.538

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
$4 | 5 x + 1 | = 3 | 2 x - 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$4 \| 5 x + 1 \| = 3 \| 2 x - 6 \|$
$x = + y$	$4 (5 x + 1) = 3 (2 x - 6)$
$x = - y$	$4 (5 x + 1) = 3 (- (2 x - 6))$
$+ x = y$	$4 (5 x + 1) = 3 (2 x - 6)$
$- x = y$	$4 (- (5 x + 1)) = 3 (2 x - 6)$

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:

$\| x \| = \| y \|$	$4 \| 5 x + 1 \| = 3 \| 2 x - 6 \|$
$x = + y$ , $+ x = y$	$4 (5 x + 1) = 3 (2 x - 6)$
$x = - y$ , $- x = y$	$4 (5 x + 1) = 3 (- (2 x - 6))$

2. Solve the two equations for x

17 additional steps

$4 \cdot (5 x + 1) = 3 \cdot (2 x - 6)$

Expand the parentheses:

$4 \cdot 5 x + 4 \cdot 1 = 3 \cdot (2 x - 6)$

Multiply the coefficients:

$20 x + 4 \cdot 1 = 3 \cdot (2 x - 6)$

Simplify the arithmetic:

$20 x + 4 = 3 \cdot (2 x - 6)$

Expand the parentheses:

$20 x + 4 = 3 \cdot 2 x + 3 \cdot - 6$

Multiply the coefficients:

$20 x + 4 = 6 x + 3 \cdot - 6$

Simplify the arithmetic:

$20 x + 4 = 6 x - 18$

Subtract from both sides:

$(20 x + 4) - 6 x = (6 x - 18) - 6 x$

Group like terms:

$(20 x - 6 x) + 4 = (6 x - 18) - 6 x$

Simplify the arithmetic:

$14 x + 4 = (6 x - 18) - 6 x$

Group like terms:

$14 x + 4 = (6 x - 6 x) - 18$

Simplify the arithmetic:

$14 x + 4 = - 18$

Subtract from both sides:

$(14 x + 4) - 4 = - 18 - 4$

Simplify the arithmetic:

$14 x = - 18 - 4$

Simplify the arithmetic:

$14 x = - 22$

Divide both sides by :

$\frac{(14 x)}{14} = \frac{- 22}{14}$

Simplify the fraction:

$x = \frac{- 22}{14}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 11 \cdot 2)}{(7 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 11}{7}$

18 additional steps

$4 \cdot (5 x + 1) = 3 \cdot (- (2 x - 6))$

Expand the parentheses:

$4 \cdot 5 x + 4 \cdot 1 = 3 \cdot (- (2 x - 6))$

Multiply the coefficients:

$20 x + 4 \cdot 1 = 3 \cdot (- (2 x - 6))$

Simplify the arithmetic:

$20 x + 4 = 3 \cdot (- (2 x - 6))$

Expand the parentheses:

$20 x + 4 = 3 \cdot (- 2 x + 6)$

Expand the parentheses:

$20 x + 4 = 3 \cdot - 2 x + 3 \cdot 6$

Multiply the coefficients:

$20 x + 4 = - 6 x + 3 \cdot 6$

Simplify the arithmetic:

$20 x + 4 = - 6 x + 18$

Add to both sides:

$(20 x + 4) + 6 x = (- 6 x + 18) + 6 x$

Group like terms:

$(20 x + 6 x) + 4 = (- 6 x + 18) + 6 x$

Simplify the arithmetic:

$26 x + 4 = (- 6 x + 18) + 6 x$

Group like terms:

$26 x + 4 = (- 6 x + 6 x) + 18$

Simplify the arithmetic:

$26 x + 4 = 18$

Subtract from both sides:

$(26 x + 4) - 4 = 18 - 4$

Simplify the arithmetic:

$26 x = 18 - 4$

Simplify the arithmetic:

$26 x = 14$

Divide both sides by :

$\frac{(26 x)}{26} = \frac{14}{26}$

Simplify the fraction:

$x = \frac{14}{26}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(7 \cdot 2)}{(13 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{7}{13}$

3. List the solutions

$x = - \frac{11}{7}, \frac{7}{13}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 4 | 5 x + 1 |$
$y = 3 | 2 x - 6 |$
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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