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Solution - Absolute value equations

Exact form:

x = \frac{28}{3}, \frac{2}{17}

x=\frac{28}{3} , \frac{2}{17}

Mixed number form:

x = 9 \frac{1}{3}, \frac{2}{17}

x=9\frac{1}{3} , \frac{2}{17}

Decimal form:

x = 9.333, 0.118

x=9.333 , 0.118

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
$| 10 x - 15 | = | 7 x + 13 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 10 x - 15 \| = \| 7 x + 13 \|$
$x = + y$	$(10 x - 15) = (7 x + 13)$
$x = - y$	$(10 x - 15) = - (7 x + 13)$
$+ x = y$	$(10 x - 15) = (7 x + 13)$
$- x = y$	$- (10 x - 15) = (7 x + 13)$

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:

$\| x \| = \| y \|$	$\| 10 x - 15 \| = \| 7 x + 13 \|$
$x = + y$ , $+ x = y$	$(10 x - 15) = (7 x + 13)$
$x = - y$ , $- x = y$	$(10 x - 15) = - (7 x + 13)$

2. Solve the two equations for x

9 additional steps

$(10 x - 15) = (7 x + 13)$

Subtract from both sides:

$(10 x - 15) - 7 x = (7 x + 13) - 7 x$

Group like terms:

$(10 x - 7 x) - 15 = (7 x + 13) - 7 x$

Simplify the arithmetic:

$3 x - 15 = (7 x + 13) - 7 x$

Group like terms:

$3 x - 15 = (7 x - 7 x) + 13$

Simplify the arithmetic:

$3 x - 15 = 13$

Add to both sides:

$(3 x - 15) + 15 = 13 + 15$

Simplify the arithmetic:

$3 x = 13 + 15$

Simplify the arithmetic:

$3 x = 28$

Divide both sides by :

$\frac{(3 x)}{3} = \frac{28}{3}$

Simplify the fraction:

$x = \frac{28}{3}$

10 additional steps

$(10 x - 15) = - (7 x + 13)$

Expand the parentheses:

$(10 x - 15) = - 7 x - 13$

Add to both sides:

$(10 x - 15) + 7 x = (- 7 x - 13) + 7 x$

Group like terms:

$(10 x + 7 x) - 15 = (- 7 x - 13) + 7 x$

Simplify the arithmetic:

$17 x - 15 = (- 7 x - 13) + 7 x$

Group like terms:

$17 x - 15 = (- 7 x + 7 x) - 13$

Simplify the arithmetic:

$17 x - 15 = - 13$

Add to both sides:

$(17 x - 15) + 15 = - 13 + 15$

Simplify the arithmetic:

$17 x = - 13 + 15$

Simplify the arithmetic:

$17 x = 2$

Divide both sides by :

$\frac{(17 x)}{17} = \frac{2}{17}$

Simplify the fraction:

$x = \frac{2}{17}$

3. List the solutions

$x = \frac{28}{3}, \frac{2}{17}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 10 x - 15 |$
$y = | 7 x + 13 |$
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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