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Solution - Absolute value equations

Exact form:

x = \frac{33}{2}

x=\frac{33}{2}

Mixed number form:

x = 16 \frac{1}{2}

x=16\frac{1}{2}

Decimal form:

x = 16.5

x=16.5

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| x - 30.4 | + | - x + 2.6 | = 0$

Add $- | - x + 2.6 |$ to both sides of the equation:

$| x - 30.4 | + | - x + 2.6 | - | - x + 2.6 | = - | - x + 2.6 |$

Simplify the arithmetic

$| x - 30.4 | = - | - x + 2.6 |$

2. 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
$| x - 30.4 | = - | - x + 2.6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x - 30.4 \| = - \| - x + 2.6 \|$
$x = + y$	$(x - 30.4) = - (- x + 2.6)$
$x = - y$	$(x - 30.4) = - - (- x + 2.6)$
$+ x = y$	$(x - 30.4) = - (- x + 2.6)$
$- x = y$	$- (x - 30.4) = - (- x + 2.6)$

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:

$\| x \| = \| y \|$	$\| x - 30.4 \| = - \| - x + 2.6 \|$
$x = + y$ , $+ x = y$	$(x - 30.4) = - (- x + 2.6)$
$x = - y$ , $- x = y$	$(x - 30.4) = - - (- x + 2.6)$

3. Solve the two equations for x

6 additional steps

$(x - 30.4) = - (- x + 2.6)$

Expand the parentheses:

$(x - 30.4) = x - 2.6$

Subtract from both sides:

$(x - 30.4) - x = (x - 2.6) - x$

Group like terms:

$(x - x) - 30.4 = (x - 2.6) - x$

Simplify the arithmetic:

$- 30.4 = (x - 2.6) - x$

Group like terms:

$- 30.4 = (x - x) - 2.6$

Simplify the arithmetic:

$- 30.4 = - 2.6$

The statement is false:

$- 30.4 = - 2.6$

The equation is false so it has no solution.

10 additional steps

$(x - 30.4) = - (- (- x + 2.6))$

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

$(x - 30.4) = - x + 2.6$

Add to both sides:

$(x - 30.4) + x = (- x + 2.6) + x$

Group like terms:

$(x + x) - 30.4 = (- x + 2.6) + x$

Simplify the arithmetic:

$2 x - 30.4 = (- x + 2.6) + x$

Group like terms:

$2 x - 30.4 = (- x + x) + 2.6$

Simplify the arithmetic:

$2 x - 30.4 = 2.6$

Add to both sides:

$(2 x - 30.4) + 30.4 = 2.6 + 30.4$

Simplify the arithmetic:

$2 x = 2.6 + 30.4$

Simplify the arithmetic:

$2 x = 33$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{33}{2}$

Simplify the fraction:

$x = \frac{33}{2}$

4. Graph

Each line represents the function of one side of the equation:
$y = | x - 30.4 |$
$y = - | - x + 2.6 |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. Graph

Why learn this

Terms and topics

Related links

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