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Solution - Absolute value equations

Exact form:

x = \frac{5}{2}

x=\frac{5}{2}

Mixed number form:

x = 2 \frac{1}{2}

x=2\frac{1}{2}

Decimal form:

x = 2.5

x=2.5

See steps

Step-by-step explanation

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

$| 7 x - 14 | - | 7 x - 21 | = 0$

Add $| 7 x - 21 |$ to both sides of the equation:

$| 7 x - 14 | - | 7 x - 21 | + | 7 x - 21 | = | 7 x - 21 |$

Simplify the arithmetic

$| 7 x - 14 | = | 7 x - 21 |$

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
$| 7 x - 14 | = | 7 x - 21 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 7 x - 14 \| = \| 7 x - 21 \|$
$x = + y$	$(7 x - 14) = (7 x - 21)$
$x = - y$	$(7 x - 14) = (- (7 x - 21))$
$+ x = y$	$(7 x - 14) = (7 x - 21)$
$- x = y$	$- (7 x - 14) = (7 x - 21)$

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:

$\| x \| = \| y \|$	$\| 7 x - 14 \| = \| 7 x - 21 \|$
$x = + y$ , $+ x = y$	$(7 x - 14) = (7 x - 21)$
$x = - y$ , $- x = y$	$(7 x - 14) = (- (7 x - 21))$

3. Solve the two equations for x

5 additional steps

$(7 x - 14) = (7 x - 21)$

Subtract from both sides:

$(7 x - 14) - 7 x = (7 x - 21) - 7 x$

Group like terms:

$(7 x - 7 x) - 14 = (7 x - 21) - 7 x$

Simplify the arithmetic:

$- 14 = (7 x - 21) - 7 x$

Group like terms:

$- 14 = (7 x - 7 x) - 21$

Simplify the arithmetic:

$- 14 = - 21$

The statement is false:

$- 14 = - 21$

The equation is false so it has no solution.

12 additional steps

$(7 x - 14) = - (7 x - 21)$

Expand the parentheses:

$(7 x - 14) = - 7 x + 21$

Add to both sides:

$(7 x - 14) + 7 x = (- 7 x + 21) + 7 x$

Group like terms:

$(7 x + 7 x) - 14 = (- 7 x + 21) + 7 x$

Simplify the arithmetic:

$14 x - 14 = (- 7 x + 21) + 7 x$

Group like terms:

$14 x - 14 = (- 7 x + 7 x) + 21$

Simplify the arithmetic:

$14 x - 14 = 21$

Add to both sides:

$(14 x - 14) + 14 = 21 + 14$

Simplify the arithmetic:

$14 x = 21 + 14$

Simplify the arithmetic:

$14 x = 35$

Divide both sides by :

$\frac{(14 x)}{14} = \frac{35}{14}$

Simplify the fraction:

$x = \frac{35}{14}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(5 \cdot 7)}{(2 \cdot 7)}$

Factor out and cancel the greatest common factor:

$x = \frac{5}{2}$

4. Graph

Each line represents the function of one side of the equation:
$y = | 7 x - 14 |$
$y = | 7 x - 21 |$
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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