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Solution - Absolute value equations

Exact form:

x = 4

x=4

See steps

Step-by-step explanation

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

$| x - 3.5 | + | - x + 4.5 | = 0$

Add $- | - x + 4.5 |$ to both sides of the equation:

$| x - 3.5 | + | - x + 4.5 | - | - x + 4.5 | = - | - x + 4.5 |$

Simplify the arithmetic

$| x - 3.5 | = - | - x + 4.5 |$

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
$| x - 3.5 | = - | - x + 4.5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x - 3.5 \| = - \| - x + 4.5 \|$
$x = + y$	$(x - 3.5) = - (- x + 4.5)$
$x = - y$	$(x - 3.5) = - - (- x + 4.5)$
$+ x = y$	$(x - 3.5) = - (- x + 4.5)$
$- x = y$	$- (x - 3.5) = - (- x + 4.5)$

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:

$\| x \| = \| y \|$	$\| x - 3.5 \| = - \| - x + 4.5 \|$
$x = + y$ , $+ x = y$	$(x - 3.5) = - (- x + 4.5)$
$x = - y$ , $- x = y$	$(x - 3.5) = - - (- x + 4.5)$

3. Solve the two equations for x

6 additional steps

$(x - 3.5) = - (- x + 4.5)$

Expand the parentheses:

$(x - 3.5) = x - 4.5$

Subtract from both sides:

$(x - 3.5) - x = (x - 4.5) - x$

Group like terms:

$(x - x) - 3.5 = (x - 4.5) - x$

Simplify the arithmetic:

$- 3.5 = (x - 4.5) - x$

Group like terms:

$- 3.5 = (x - x) - 4.5$

Simplify the arithmetic:

$- 3.5 = - 4.5$

The statement is false:

$- 3.5 = - 4.5$

The equation is false so it has no solution.

12 additional steps

$(x - 3.5) = - (- (- x + 4.5))$

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

$(x - 3.5) = - x + 4.5$

Add to both sides:

$(x - 3.5) + x = (- x + 4.5) + x$

Group like terms:

$(x + x) - 3.5 = (- x + 4.5) + x$

Simplify the arithmetic:

$2 x - 3.5 = (- x + 4.5) + x$

Group like terms:

$2 x - 3.5 = (- x + x) + 4.5$

Simplify the arithmetic:

$2 x - 3.5 = 4.5$

Add to both sides:

$(2 x - 3.5) + 3.5 = 4.5 + 3.5$

Simplify the arithmetic:

$2 x = 4.5 + 3.5$

Simplify the arithmetic:

$2 x = 8$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{8}{2}$

Simplify the fraction:

$x = \frac{8}{2}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(4 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = 4$

4. Graph

Each line represents the function of one side of the equation:
$y = | x - 3.5 |$
$y = - | - x + 4.5 |$
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

Latest Related Drills Solved