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Solution - Absolute value equations

Exact form:

x = \frac{15}{2}, - \frac{1}{2}

x=\frac{15}{2} , -\frac{1}{2}

Mixed number form:

x = 7 \frac{1}{2}, - \frac{1}{2}

x=7\frac{1}{2} , -\frac{1}{2}

Decimal form:

x = 7.5, - 0.5

x=7.5 , -0.5

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
$| 10 x - 11 | = | 6 x + 19 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 10 x - 11 \| = \| 6 x + 19 \|$
$x = + y$	$(10 x - 11) = (6 x + 19)$
$x = - y$	$(10 x - 11) = - (6 x + 19)$
$+ x = y$	$(10 x - 11) = (6 x + 19)$
$- x = y$	$- (10 x - 11) = (6 x + 19)$

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:

$\| x \| = \| y \|$	$\| 10 x - 11 \| = \| 6 x + 19 \|$
$x = + y$ , $+ x = y$	$(10 x - 11) = (6 x + 19)$
$x = - y$ , $- x = y$	$(10 x - 11) = - (6 x + 19)$

2. Solve the two equations for x

11 additional steps

$(10 x - 11) = (6 x + 19)$

Subtract from both sides:

$(10 x - 11) - 6 x = (6 x + 19) - 6 x$

Group like terms:

$(10 x - 6 x) - 11 = (6 x + 19) - 6 x$

Simplify the arithmetic:

$4 x - 11 = (6 x + 19) - 6 x$

Group like terms:

$4 x - 11 = (6 x - 6 x) + 19$

Simplify the arithmetic:

$4 x - 11 = 19$

Add to both sides:

$(4 x - 11) + 11 = 19 + 11$

Simplify the arithmetic:

$4 x = 19 + 11$

Simplify the arithmetic:

$4 x = 30$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{30}{4}$

Simplify the fraction:

$x = \frac{30}{4}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(15 \cdot 2)}{(2 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{15}{2}$

12 additional steps

$(10 x - 11) = - (6 x + 19)$

Expand the parentheses:

$(10 x - 11) = - 6 x - 19$

Add to both sides:

$(10 x - 11) + 6 x = (- 6 x - 19) + 6 x$

Group like terms:

$(10 x + 6 x) - 11 = (- 6 x - 19) + 6 x$

Simplify the arithmetic:

$16 x - 11 = (- 6 x - 19) + 6 x$

Group like terms:

$16 x - 11 = (- 6 x + 6 x) - 19$

Simplify the arithmetic:

$16 x - 11 = - 19$

Add to both sides:

$(16 x - 11) + 11 = - 19 + 11$

Simplify the arithmetic:

$16 x = - 19 + 11$

Simplify the arithmetic:

$16 x = - 8$

Divide both sides by :

$\frac{(16 x)}{16} = \frac{- 8}{16}$

Simplify the fraction:

$x = \frac{- 8}{16}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 1 \cdot 8)}{(2 \cdot 8)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 1}{2}$

3. List the solutions

$x = \frac{15}{2}, - \frac{1}{2}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 10 x - 11 |$
$y = | 6 x + 19 |$
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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