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Solution - Absolute value equations

Exact form:

x = - 5, \frac{5}{6}

x=-5 , \frac{5}{6}

Decimal form:

x = - 5, 0.833

x=-5 , 0.833

See steps

Step-by-step explanation

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

$7 | x | - 5 | x - 2 | = 0$

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

$7 | x | - 5 | x - 2 | + 5 | x - 2 | = 5 | x - 2 |$

Simplify the arithmetic

$7 | x | = 5 | x - 2 |$

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
$7 | x | = 5 | x - 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$7 \| x \| = 5 \| x - 2 \|$
$x = + y$	$7 (x) = 5 (x - 2)$
$x = - y$	$7 (x) = 5 (- (x - 2))$
$+ x = y$	$7 (x) = 5 (x - 2)$
$- x = y$	$7 (- (x)) = 5 (x - 2)$

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:

$\| x \| = \| y \|$	$7 \| x \| = 5 \| x - 2 \|$
$x = + y$ , $+ x = y$	$7 (x) = 5 (x - 2)$
$x = - y$ , $- x = y$	$7 (x) = 5 (- (x - 2))$

3. Solve the two equations for x

9 additional steps

$7 x = 5 \cdot (x - 2)$

Expand the parentheses:

$7 x = 5 x + 5 \cdot - 2$

Simplify the arithmetic:

$7 x = 5 x - 10$

Subtract from both sides:

$(7 x) - 5 x = (5 x - 10) - 5 x$

Simplify the arithmetic:

$2 x = (5 x - 10) - 5 x$

Group like terms:

$2 x = (5 x - 5 x) - 10$

Simplify the arithmetic:

$2 x = - 10$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{- 10}{2}$

Simplify the fraction:

$x = \frac{- 10}{2}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 5 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = - 5$

12 additional steps

$7 x = 5 \cdot (- (x - 2))$

Expand the parentheses:

$7 x = 5 \cdot (- x + 2)$

$7 x = 5 \cdot - x + 5 \cdot 2$

Group like terms:

$7 x = (5 \cdot - 1) x + 5 \cdot 2$

Multiply the coefficients:

$7 x = - 5 x + 5 \cdot 2$

Simplify the arithmetic:

$7 x = - 5 x + 10$

Add to both sides:

$(7 x) + 5 x = (- 5 x + 10) + 5 x$

Simplify the arithmetic:

$12 x = (- 5 x + 10) + 5 x$

Group like terms:

$12 x = (- 5 x + 5 x) + 10$

Simplify the arithmetic:

$12 x = 10$

Divide both sides by :

$\frac{(12 x)}{12} = \frac{10}{12}$

Simplify the fraction:

$x = \frac{10}{12}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(5 \cdot 2)}{(6 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{5}{6}$

4. List the solutions

$x = - 5, \frac{5}{6}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = 7 | x |$
$y = 5 | x - 2 |$
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. List the solutions

5. 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. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved