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Solution - Absolute value equations

Exact form:

x = - 1, \frac{5}{11}

x=-1 , \frac{5}{11}

Decimal form:

x = - 1, 0.455

x=-1 , 0.455

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
$| x - 7 | = 2 | 5 x + 1 |$
without the absolute value bars:

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

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:

$\| x \| = \| y \|$	$\| x - 7 \| = 2 \| 5 x + 1 \|$
$x = + y$ , $+ x = y$	$(x - 7) = 2 (5 x + 1)$
$x = - y$ , $- x = y$	$(x - 7) = 2 (- (5 x + 1))$

2. Solve the two equations for x

15 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

$(x - 7) = 10 x + 2 \cdot 1$

Simplify the arithmetic:

$(x - 7) = 10 x + 2$

Subtract from both sides:

$(x - 7) - 10 x = (10 x + 2) - 10 x$

Group like terms:

$(x - 10 x) - 7 = (10 x + 2) - 10 x$

Simplify the arithmetic:

$- 9 x - 7 = (10 x + 2) - 10 x$

Group like terms:

$- 9 x - 7 = (10 x - 10 x) + 2$

Simplify the arithmetic:

$- 9 x - 7 = 2$

Add to both sides:

$(- 9 x - 7) + 7 = 2 + 7$

Simplify the arithmetic:

$- 9 x = 2 + 7$

Simplify the arithmetic:

$- 9 x = 9$

Divide both sides by :

$\frac{(- 9 x)}{- 9} = \frac{9}{- 9}$

Cancel out the negatives:

$\frac{9 x}{9} = \frac{9}{- 9}$

Simplify the fraction:

$x = \frac{9}{- 9}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 9}{9}$

Simplify the fraction:

$x = - 1$

13 additional steps

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

Expand the parentheses:

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

Expand the parentheses:

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

Multiply the coefficients:

$(x - 7) = - 10 x + 2 \cdot - 1$

Simplify the arithmetic:

$(x - 7) = - 10 x - 2$

Add to both sides:

$(x - 7) + 10 x = (- 10 x - 2) + 10 x$

Group like terms:

$(x + 10 x) - 7 = (- 10 x - 2) + 10 x$

Simplify the arithmetic:

$11 x - 7 = (- 10 x - 2) + 10 x$

Group like terms:

$11 x - 7 = (- 10 x + 10 x) - 2$

Simplify the arithmetic:

$11 x - 7 = - 2$

Add to both sides:

$(11 x - 7) + 7 = - 2 + 7$

Simplify the arithmetic:

$11 x = - 2 + 7$

Simplify the arithmetic:

$11 x = 5$

Divide both sides by :

$\frac{(11 x)}{11} = \frac{5}{11}$

Simplify the fraction:

$x = \frac{5}{11}$

3. List the solutions

$x = - 1, \frac{5}{11}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | x - 7 |$
$y = 2 | 5 x + 1 |$
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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