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Solution - Absolute value equations

Exact form:

x = \frac{3}{2}

x=\frac{3}{2}

Mixed number form:

x = 1 \frac{1}{2}

x=1\frac{1}{2}

Decimal form:

x = 1.5

x=1.5

See steps

Step-by-step explanation

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

$| 2 x - 1 | - | 2 x - 5 | = 0$

Add $| 2 x - 5 |$ to both sides of the equation:

$| 2 x - 1 | - | 2 x - 5 | + | 2 x - 5 | = | 2 x - 5 |$

Simplify the arithmetic

$| 2 x - 1 | = | 2 x - 5 |$

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
$| 2 x - 1 | = | 2 x - 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 x - 1 \| = \| 2 x - 5 \|$
$x = + y$	$(2 x - 1) = (2 x - 5)$
$x = - y$	$(2 x - 1) = (- (2 x - 5))$
$+ x = y$	$(2 x - 1) = (2 x - 5)$
$- x = y$	$- (2 x - 1) = (2 x - 5)$

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:

$\| x \| = \| y \|$	$\| 2 x - 1 \| = \| 2 x - 5 \|$
$x = + y$ , $+ x = y$	$(2 x - 1) = (2 x - 5)$
$x = - y$ , $- x = y$	$(2 x - 1) = (- (2 x - 5))$

3. Solve the two equations for x

5 additional steps

$(2 x - 1) = (2 x - 5)$

Subtract from both sides:

$(2 x - 1) - 2 x = (2 x - 5) - 2 x$

Group like terms:

$(2 x - 2 x) - 1 = (2 x - 5) - 2 x$

Simplify the arithmetic:

$- 1 = (2 x - 5) - 2 x$

Group like terms:

$- 1 = (2 x - 2 x) - 5$

Simplify the arithmetic:

$- 1 = - 5$

The statement is false:

$- 1 = - 5$

The equation is false so it has no solution.

12 additional steps

$(2 x - 1) = - (2 x - 5)$

Expand the parentheses:

$(2 x - 1) = - 2 x + 5$

Add to both sides:

$(2 x - 1) + 2 x = (- 2 x + 5) + 2 x$

Group like terms:

$(2 x + 2 x) - 1 = (- 2 x + 5) + 2 x$

Simplify the arithmetic:

$4 x - 1 = (- 2 x + 5) + 2 x$

Group like terms:

$4 x - 1 = (- 2 x + 2 x) + 5$

Simplify the arithmetic:

$4 x - 1 = 5$

Add to both sides:

$(4 x - 1) + 1 = 5 + 1$

Simplify the arithmetic:

$4 x = 5 + 1$

Simplify the arithmetic:

$4 x = 6$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{6}{4}$

Simplify the fraction:

$x = \frac{6}{4}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(3 \cdot 2)}{(2 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{3}{2}$

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 x - 1 |$
$y = | 2 x - 5 |$
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

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