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Solution - Absolute value equations

Exact form:

x = - \frac{3}{10}

x=-\frac{3}{10}

Decimal form:

x = - 0.3

x=-0.3

See steps

Step-by-step explanation

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

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

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:

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

2. Solve the two equations for x

13 additional steps

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

Expand the parentheses:

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

Simplify the arithmetic:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 10 x + 2 = 5$

Subtract from both sides:

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

Simplify the arithmetic:

$- 10 x = 5 - 2$

Simplify the arithmetic:

$- 10 x = 3$

Divide both sides by :

$\frac{(- 10 x)}{- 10} = \frac{3}{- 10}$

Cancel out the negatives:

$\frac{10 x}{10} = \frac{3}{- 10}$

Simplify the fraction:

$x = \frac{3}{- 10}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 3}{10}$

10 additional steps

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

Expand the parentheses:

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

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

Group like terms:

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

Multiply the coefficients:

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

Simplify the arithmetic:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$2 = - 5$

The statement is false:

$2 = - 5$

The equation is false so it has no solution.

3. List the solutions

$x = - \frac{3}{10}$
(1 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 5 x + 2 |$
$y = 5 | x + 1 |$
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 without absolute value bars

2. Solve the two equations for x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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