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Solution - Absolute value equations

Exact form:

x = \frac{16}{9}, \frac{8}{5}

x=\frac{16}{9} , \frac{8}{5}

Mixed number form:

x = 1 \frac{7}{9}, 1 \frac{3}{5}

x=1\frac{7}{9} , 1\frac{3}{5}

Decimal form:

x = 1.778, 1.6

x=1.778 , 1.6

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

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

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:

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

2. Solve the two equations for x

12 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

$12 x + 4 \cdot - 5 = (3 x - 4)$

Simplify the arithmetic:

$12 x - 20 = (3 x - 4)$

Subtract from both sides:

$(12 x - 20) - 3 x = (3 x - 4) - 3 x$

Group like terms:

$(12 x - 3 x) - 20 = (3 x - 4) - 3 x$

Simplify the arithmetic:

$9 x - 20 = (3 x - 4) - 3 x$

Group like terms:

$9 x - 20 = (3 x - 3 x) - 4$

Simplify the arithmetic:

$9 x - 20 = - 4$

Add to both sides:

$(9 x - 20) + 20 = - 4 + 20$

Simplify the arithmetic:

$9 x = - 4 + 20$

Simplify the arithmetic:

$9 x = 16$

Divide both sides by :

$\frac{(9 x)}{9} = \frac{16}{9}$

Simplify the fraction:

$x = \frac{16}{9}$

15 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

$12 x + 4 \cdot - 5 = - (3 x - 4)$

Simplify the arithmetic:

$12 x - 20 = - (3 x - 4)$

Expand the parentheses:

$12 x - 20 = - 3 x + 4$

Add to both sides:

$(12 x - 20) + 3 x = (- 3 x + 4) + 3 x$

Group like terms:

$(12 x + 3 x) - 20 = (- 3 x + 4) + 3 x$

Simplify the arithmetic:

$15 x - 20 = (- 3 x + 4) + 3 x$

Group like terms:

$15 x - 20 = (- 3 x + 3 x) + 4$

Simplify the arithmetic:

$15 x - 20 = 4$

Add to both sides:

$(15 x - 20) + 20 = 4 + 20$

Simplify the arithmetic:

$15 x = 4 + 20$

Simplify the arithmetic:

$15 x = 24$

Divide both sides by :

$\frac{(15 x)}{15} = \frac{24}{15}$

Simplify the fraction:

$x = \frac{24}{15}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(8 \cdot 3)}{(5 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = \frac{8}{5}$

3. List the solutions

$x = \frac{16}{9}, \frac{8}{5}$
(2 solution(s))

4. Graph

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