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Solution - Absolute value equations

Exact form:

x = \frac{3}{8}, \frac{45}{14}

x=\frac{3}{8} , \frac{45}{14}

Mixed number form:

x = \frac{3}{8}, 3 \frac{3}{14}

x=\frac{3}{8} , 3\frac{3}{14}

Decimal form:

x = 0.375, 3.214

x=0.375 , 3.214

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
$| - \frac{11}{3} x + 8 | = | - x + 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - \frac{11}{3} x + 8 \| = \| - x + 7 \|$
$x = + y$	$(- \frac{11}{3} x + 8) = (- x + 7)$
$x = - y$	$(- \frac{11}{3} x + 8) = - (- x + 7)$
$+ x = y$	$(- \frac{11}{3} x + 8) = (- x + 7)$
$- x = y$	$- (- \frac{11}{3} x + 8) = (- x + 7)$

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:

$\| x \| = \| y \|$	$\| - \frac{11}{3} x + 8 \| = \| - x + 7 \|$
$x = + y$ , $+ x = y$	$(- \frac{11}{3} x + 8) = (- x + 7)$
$x = - y$ , $- x = y$	$(- \frac{11}{3} x + 8) = - (- x + 7)$

2. Solve the two equations for x

17 additional steps

$(\frac{- 11}{3} x + 8) = (- x + 7)$

Add to both sides:

$(\frac{- 11}{3} x + 8) + x = (- x + 7) + x$

Group like terms:

$(\frac{- 11}{3} x + x) + 8 = (- x + 7) + x$

Group the coefficients:

$(\frac{- 11}{3} + 1) x + 8 = (- x + 7) + x$

Convert the integer into a fraction:

$(\frac{- 11}{3} + \frac{3}{3}) x + 8 = (- x + 7) + x$

Combine the fractions:

$\frac{(- 11 + 3)}{3} x + 8 = (- x + 7) + x$

Combine the numerators:

$\frac{- 8}{3} x + 8 = (- x + 7) + x$

Group like terms:

$\frac{- 8}{3} x + 8 = (- x + x) + 7$

Simplify the arithmetic:

$\frac{- 8}{3} x + 8 = 7$

Subtract from both sides:

$(\frac{- 8}{3} x + 8) - 8 = 7 - 8$

Simplify the arithmetic:

$\frac{- 8}{3} x = 7 - 8$

Simplify the arithmetic:

$\frac{- 8}{3} x = - 1$

Multiply both sides by inverse fraction :

$(\frac{- 8}{3} x) \cdot \frac{3}{- 8} = - 1 \cdot \frac{3}{- 8}$

Move the negative sign from the denominator to the numerator:

$\frac{- 8}{3} x \cdot \frac{- 3}{8} = - 1 \cdot \frac{3}{- 8}$

Group like terms:

$(\frac{- 8}{3} \cdot \frac{- 3}{8}) x = - 1 \cdot \frac{3}{- 8}$

Multiply the coefficients:

$\frac{(- 8 \cdot - 3)}{(3 \cdot 8)} x = - 1 \cdot \frac{3}{- 8}$

Simplify the arithmetic:

$1 x = - 1 \cdot \frac{3}{- 8}$

$x = - 1 \cdot \frac{3}{- 8}$

Cancel out the negatives:

$x = \frac{3}{8}$

20 additional steps

$(\frac{- 11}{3} x + 8) = - (- x + 7)$

Expand the parentheses:

$(\frac{- 11}{3} x + 8) = x - 7$

Subtract from both sides:

$(\frac{- 11}{3} x + 8) - x = (x - 7) - x$

Group like terms:

$(\frac{- 11}{3} x - x) + 8 = (x - 7) - x$

Group the coefficients:

$(\frac{- 11}{3} - 1) x + 8 = (x - 7) - x$

Convert the integer into a fraction:

$(\frac{- 11}{3} + \frac{- 3}{3}) x + 8 = (x - 7) - x$

Combine the fractions:

$\frac{(- 11 - 3)}{3} x + 8 = (x - 7) - x$

Combine the numerators:

$\frac{- 14}{3} x + 8 = (x - 7) - x$

Group like terms:

$\frac{- 14}{3} x + 8 = (x - x) - 7$

Simplify the arithmetic:

$\frac{- 14}{3} x + 8 = - 7$

Subtract from both sides:

$(\frac{- 14}{3} x + 8) - 8 = - 7 - 8$

Simplify the arithmetic:

$\frac{- 14}{3} x = - 7 - 8$

Simplify the arithmetic:

$\frac{- 14}{3} x = - 15$

Multiply both sides by inverse fraction :

$(\frac{- 14}{3} x) \cdot \frac{3}{- 14} = - 15 \cdot \frac{3}{- 14}$

Move the negative sign from the denominator to the numerator:

$\frac{- 14}{3} x \cdot \frac{- 3}{14} = - 15 \cdot \frac{3}{- 14}$

Group like terms:

$(\frac{- 14}{3} \cdot \frac{- 3}{14}) x = - 15 \cdot \frac{3}{- 14}$

Multiply the coefficients:

$\frac{(- 14 \cdot - 3)}{(3 \cdot 14)} x = - 15 \cdot \frac{3}{- 14}$

Simplify the arithmetic:

$1 x = - 15 \cdot \frac{3}{- 14}$

$x = - 15 \cdot \frac{3}{- 14}$

Move the negative sign from the denominator to the numerator:

$x = - 15 \cdot \frac{- 3}{14}$

Multiply the fraction(s):

$x = \frac{(- 15 \cdot - 3)}{14}$

Simplify the arithmetic:

$x = \frac{45}{14}$

3. List the solutions

$x = \frac{3}{8}, \frac{45}{14}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - \frac{11}{3} x + 8 |$
$y = | - x + 7 |$
The equation is true where the two lines cross.

How did we do?

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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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