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Solution - Absolute value equations

Exact form:

x = \frac{5}{3}, \frac{15}{11}

x=\frac{5}{3} , \frac{15}{11}

Mixed number form:

x = 1 \frac{2}{3}, 1 \frac{4}{11}

x=1\frac{2}{3} , 1\frac{4}{11}

Decimal form:

x = 1.667, 1.364

x=1.667 , 1.364

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
$5 | 2 x - 3 | = | x |$
without the absolute value bars:

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

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:

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

2. Solve the two equations for x

13 additional steps

$5 \cdot (2 x - 3) = x$

Expand the parentheses:

$5 \cdot 2 x + 5 \cdot - 3 = x$

Multiply the coefficients:

$10 x + 5 \cdot - 3 = x$

Simplify the arithmetic:

$10 x - 15 = x$

Subtract from both sides:

$(10 x - 15) - x = x - x$

Group like terms:

$(10 x - x) - 15 = x - x$

Simplify the arithmetic:

$9 x - 15 = x - x$

Simplify the arithmetic:

$9 x - 15 = 0$

Add to both sides:

$(9 x - 15) + 15 = 0 + 15$

Simplify the arithmetic:

$9 x = 0 + 15$

Simplify the arithmetic:

$9 x = 15$

Divide both sides by :

$\frac{(9 x)}{9} = \frac{15}{9}$

Simplify the fraction:

$x = \frac{15}{9}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$x = \frac{5}{3}$

11 additional steps

$5 \cdot (2 x - 3) = - x$

Expand the parentheses:

$5 \cdot 2 x + 5 \cdot - 3 = - x$

Multiply the coefficients:

$10 x + 5 \cdot - 3 = - x$

Simplify the arithmetic:

$10 x - 15 = - x$

Add to both sides:

$(10 x - 15) + x = - x + x$

Group like terms:

$(10 x + x) - 15 = - x + x$

Simplify the arithmetic:

$11 x - 15 = - x + x$

Simplify the arithmetic:

$11 x - 15 = 0$

Add to both sides:

$(11 x - 15) + 15 = 0 + 15$

Simplify the arithmetic:

$11 x = 0 + 15$

Simplify the arithmetic:

$11 x = 15$

Divide both sides by :

$\frac{(11 x)}{11} = \frac{15}{11}$

Simplify the fraction:

$x = \frac{15}{11}$

3. List the solutions

$x = \frac{5}{3}, \frac{15}{11}$
(2 solution(s))

4. Graph

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