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Solution - Absolute value equations

Exact form:

x = \frac{19}{3}, \frac{3}{17}

x=\frac{19}{3} , \frac{3}{17}

Mixed number form:

x = 6 \frac{1}{3}, \frac{3}{17}

x=6\frac{1}{3} , \frac{3}{17}

Decimal form:

x = 6.333, 0.176

x=6.333 , 0.176

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
$| 10 x - 11 | = | 7 x + 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 10 x - 11 \| = \| 7 x + 8 \|$
$x = + y$	$(10 x - 11) = (7 x + 8)$
$x = - y$	$(10 x - 11) = - (7 x + 8)$
$+ x = y$	$(10 x - 11) = (7 x + 8)$
$- x = y$	$- (10 x - 11) = (7 x + 8)$

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:

$\| x \| = \| y \|$	$\| 10 x - 11 \| = \| 7 x + 8 \|$
$x = + y$ , $+ x = y$	$(10 x - 11) = (7 x + 8)$
$x = - y$ , $- x = y$	$(10 x - 11) = - (7 x + 8)$

2. Solve the two equations for x

9 additional steps

$(10 x - 11) = (7 x + 8)$

Subtract from both sides:

$(10 x - 11) - 7 x = (7 x + 8) - 7 x$

Group like terms:

$(10 x - 7 x) - 11 = (7 x + 8) - 7 x$

Simplify the arithmetic:

$3 x - 11 = (7 x + 8) - 7 x$

Group like terms:

$3 x - 11 = (7 x - 7 x) + 8$

Simplify the arithmetic:

$3 x - 11 = 8$

Add to both sides:

$(3 x - 11) + 11 = 8 + 11$

Simplify the arithmetic:

$3 x = 8 + 11$

Simplify the arithmetic:

$3 x = 19$

Divide both sides by :

$\frac{(3 x)}{3} = \frac{19}{3}$

Simplify the fraction:

$x = \frac{19}{3}$

10 additional steps

$(10 x - 11) = - (7 x + 8)$

Expand the parentheses:

$(10 x - 11) = - 7 x - 8$

Add to both sides:

$(10 x - 11) + 7 x = (- 7 x - 8) + 7 x$

Group like terms:

$(10 x + 7 x) - 11 = (- 7 x - 8) + 7 x$

Simplify the arithmetic:

$17 x - 11 = (- 7 x - 8) + 7 x$

Group like terms:

$17 x - 11 = (- 7 x + 7 x) - 8$

Simplify the arithmetic:

$17 x - 11 = - 8$

Add to both sides:

$(17 x - 11) + 11 = - 8 + 11$

Simplify the arithmetic:

$17 x = - 8 + 11$

Simplify the arithmetic:

$17 x = 3$

Divide both sides by :

$\frac{(17 x)}{17} = \frac{3}{17}$

Simplify the fraction:

$x = \frac{3}{17}$

3. List the solutions

$x = \frac{19}{3}, \frac{3}{17}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 10 x - 11 |$
$y = | 7 x + 8 |$
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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