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Solution - Absolute value equations

Exact form:

g = \frac{5}{2}, - 40

g=\frac{5}{2} , -40

Mixed number form:

g = 2 \frac{1}{2}, - 40

g=2\frac{1}{2} , -40

Decimal form:

g = 2.5, - 40

g=2.5 , -40

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
$| 16 g - 40 | = - 4 | 4 g - 10 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 16 g - 40 \| = - 4 \| 4 g - 10 \|$
$x = + y$	$(16 g - 40) = - 4 (4 g - 10)$
$x = - y$	$(16 g - 40) = - 4 (- (4 g - 10))$
$+ x = y$	$(16 g - 40) = - 4 (4 g - 10)$
$- x = y$	$- (16 g - 40) = - 4 (4 g - 10)$

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:

$\| x \| = \| y \|$	$\| 16 g - 40 \| = - 4 \| 4 g - 10 \|$
$x = + y$ , $+ x = y$	$(16 g - 40) = - 4 (4 g - 10)$
$x = - y$ , $- x = y$	$(16 g - 40) = - 4 (- (4 g - 10))$

2. Solve the two equations for g

14 additional steps

$(16 g - 40) = - 4 \cdot (4 g - 10)$

Expand the parentheses:

$(16 g - 40) = - 4 \cdot 4 g - 4 \cdot - 10$

Multiply the coefficients:

$(16 g - 40) = - 16 g - 4 \cdot - 10$

Simplify the arithmetic:

$(16 g - 40) = - 16 g + 40$

Add to both sides:

$(16 g - 40) + 16 g = (- 16 g + 40) + 16 g$

Group like terms:

$(16 g + 16 g) - 40 = (- 16 g + 40) + 16 g$

Simplify the arithmetic:

$32 g - 40 = (- 16 g + 40) + 16 g$

Group like terms:

$32 g - 40 = (- 16 g + 16 g) + 40$

Simplify the arithmetic:

$32 g - 40 = 40$

Add to both sides:

$(32 g - 40) + 40 = 40 + 40$

Simplify the arithmetic:

$32 g = 40 + 40$

Simplify the arithmetic:

$32 g = 80$

Divide both sides by :

$\frac{(32 g)}{32} = \frac{80}{32}$

Simplify the fraction:

$g = \frac{80}{32}$

Find the greatest common factor of the numerator and denominator:

$g = \frac{(5 \cdot 16)}{(2 \cdot 16)}$

Factor out and cancel the greatest common factor:

$g = \frac{5}{2}$

8 additional steps

$(16 g - 40) = - 4 \cdot (- (4 g - 10))$

Expand the parentheses:

$(16 g - 40) = - 4 \cdot (- 4 g + 10)$

Expand the parentheses:

$(16 g - 40) = - 4 \cdot - 4 g - 4 \cdot 10$

Multiply the coefficients:

$(16 g - 40) = 16 g - 4 \cdot 10$

Simplify the arithmetic:

$(16 g - 40) = 16 g - 40$

Subtract from both sides:

$(16 g - 40) - 16 g = (16 g - 40) - 16 g$

Group like terms:

$(16 g - 16 g) - 40 = (16 g - 40) - 16 g$

Simplify the arithmetic:

$- 40 = (16 g - 40) - 16 g$

Group like terms:

$- 40 = (16 g - 16 g) - 40$

Simplify the arithmetic:

$- 40 = - 40$

3. List the solutions

$g = \frac{5}{2}, - 40$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 16 g - 40 |$
$y = - 4 | 4 g - 10 |$
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 g

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 g

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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