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Solution - Absolute value equations

Exact form:

x = \frac{17}{9}, \frac{13}{11}

x=\frac{17}{9} , \frac{13}{11}

Mixed number form:

x = 1 \frac{8}{9}, 1 \frac{2}{11}

x=1\frac{8}{9} , 1\frac{2}{11}

Decimal form:

x = 1.889, 1.182

x=1.889 , 1.182

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

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

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:

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

2. Solve the two equations for x

12 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

$10 x + 5 \cdot - 3 = (x + 2)$

Simplify the arithmetic:

$10 x - 15 = (x + 2)$

Subtract from both sides:

$(10 x - 15) - x = (x + 2) - x$

Group like terms:

$(10 x - x) - 15 = (x + 2) - x$

Simplify the arithmetic:

$9 x - 15 = (x + 2) - x$

Group like terms:

$9 x - 15 = (x - x) + 2$

Simplify the arithmetic:

$9 x - 15 = 2$

Add to both sides:

$(9 x - 15) + 15 = 2 + 15$

Simplify the arithmetic:

$9 x = 2 + 15$

Simplify the arithmetic:

$9 x = 17$

Divide both sides by :

$\frac{(9 x)}{9} = \frac{17}{9}$

Simplify the fraction:

$x = \frac{17}{9}$

13 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

$10 x + 5 \cdot - 3 = - (x + 2)$

Simplify the arithmetic:

$10 x - 15 = - (x + 2)$

Expand the parentheses:

$10 x - 15 = - x - 2$

Add to both sides:

$(10 x - 15) + x = (- x - 2) + x$

Group like terms:

$(10 x + x) - 15 = (- x - 2) + x$

Simplify the arithmetic:

$11 x - 15 = (- x - 2) + x$

Group like terms:

$11 x - 15 = (- x + x) - 2$

Simplify the arithmetic:

$11 x - 15 = - 2$

Add to both sides:

$(11 x - 15) + 15 = - 2 + 15$

Simplify the arithmetic:

$11 x = - 2 + 15$

Simplify the arithmetic:

$11 x = 13$

Divide both sides by :

$\frac{(11 x)}{11} = \frac{13}{11}$

Simplify the fraction:

$x = \frac{13}{11}$

3. List the solutions

$x = \frac{17}{9}, \frac{13}{11}$
(2 solution(s))

4. Graph

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