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Solution - Absolute value equations

Exact form:

x = - 29, \frac{13}{15}

x=-29 , \frac{13}{15}

Decimal form:

x = - 29, 0.867

x=-29 , 0.867

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
$4 | 2 x + 2 | = 7 | x - 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$4 \| 2 x + 2 \| = 7 \| x - 3 \|$
$x = + y$	$4 (2 x + 2) = 7 (x - 3)$
$x = - y$	$4 (2 x + 2) = 7 (- (x - 3))$
$+ x = y$	$4 (2 x + 2) = 7 (x - 3)$
$- x = y$	$4 (- (2 x + 2)) = 7 (x - 3)$

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:

$\| x \| = \| y \|$	$4 \| 2 x + 2 \| = 7 \| x - 3 \|$
$x = + y$ , $+ x = y$	$4 (2 x + 2) = 7 (x - 3)$
$x = - y$ , $- x = y$	$4 (2 x + 2) = 7 (- (x - 3))$

2. Solve the two equations for x

12 additional steps

$4 \cdot (2 x + 2) = 7 \cdot (x - 3)$

Expand the parentheses:

$4 \cdot 2 x + 4 \cdot 2 = 7 \cdot (x - 3)$

Multiply the coefficients:

$8 x + 4 \cdot 2 = 7 \cdot (x - 3)$

Simplify the arithmetic:

$8 x + 8 = 7 \cdot (x - 3)$

Expand the parentheses:

$8 x + 8 = 7 x + 7 \cdot - 3$

Simplify the arithmetic:

$8 x + 8 = 7 x - 21$

Subtract from both sides:

$(8 x + 8) - 7 x = (7 x - 21) - 7 x$

Group like terms:

$(8 x - 7 x) + 8 = (7 x - 21) - 7 x$

Simplify the arithmetic:

$x + 8 = (7 x - 21) - 7 x$

Group like terms:

$x + 8 = (7 x - 7 x) - 21$

Simplify the arithmetic:

$x + 8 = - 21$

Subtract from both sides:

$(x + 8) - 8 = - 21 - 8$

Simplify the arithmetic:

$x = - 21 - 8$

Simplify the arithmetic:

$x = - 29$

17 additional steps

$4 \cdot (2 x + 2) = 7 \cdot (- (x - 3))$

Expand the parentheses:

$4 \cdot 2 x + 4 \cdot 2 = 7 \cdot (- (x - 3))$

Multiply the coefficients:

$8 x + 4 \cdot 2 = 7 \cdot (- (x - 3))$

Simplify the arithmetic:

$8 x + 8 = 7 \cdot (- (x - 3))$

Expand the parentheses:

$8 x + 8 = 7 \cdot (- x + 3)$

$8 x + 8 = 7 \cdot - x + 7 \cdot 3$

Group like terms:

$8 x + 8 = (7 \cdot - 1) x + 7 \cdot 3$

Multiply the coefficients:

$8 x + 8 = - 7 x + 7 \cdot 3$

Simplify the arithmetic:

$8 x + 8 = - 7 x + 21$

Add to both sides:

$(8 x + 8) + 7 x = (- 7 x + 21) + 7 x$

Group like terms:

$(8 x + 7 x) + 8 = (- 7 x + 21) + 7 x$

Simplify the arithmetic:

$15 x + 8 = (- 7 x + 21) + 7 x$

Group like terms:

$15 x + 8 = (- 7 x + 7 x) + 21$

Simplify the arithmetic:

$15 x + 8 = 21$

Subtract from both sides:

$(15 x + 8) - 8 = 21 - 8$

Simplify the arithmetic:

$15 x = 21 - 8$

Simplify the arithmetic:

$15 x = 13$

Divide both sides by :

$\frac{(15 x)}{15} = \frac{13}{15}$

Simplify the fraction:

$x = \frac{13}{15}$

3. List the solutions

$x = - 29, \frac{13}{15}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 4 | 2 x + 2 |$
$y = 7 | x - 3 |$
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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