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Solution - Absolute value equations

Exact form:

x = - \frac{13}{2}

x=-\frac{13}{2}

Mixed number form:

x = - 6 \frac{1}{2}

x=-6\frac{1}{2}

Decimal form:

x = - 6.5

x=-6.5

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

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

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:

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

2. Solve the two equations for x

5 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

$19 = (x - 6) - x$

Group like terms:

$19 = (x - x) - 6$

Simplify the arithmetic:

$19 = - 6$

The statement is false:

$19 = - 6$

The equation is false so it has no solution.

10 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$2 x + 19 = (- x + 6) + x$

Group like terms:

$2 x + 19 = (- x + x) + 6$

Simplify the arithmetic:

$2 x + 19 = 6$

Subtract from both sides:

$(2 x + 19) - 19 = 6 - 19$

Simplify the arithmetic:

$2 x = 6 - 19$

Simplify the arithmetic:

$2 x = - 13$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{- 13}{2}$

Simplify the fraction:

$x = \frac{- 13}{2}$

3. Graph

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

Why learn this

Terms and topics

Related links

Latest Related Drills Solved