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Solution - Absolute value equations

Exact form:

h = 4, - \frac{2}{3}

h=4 , -\frac{2}{3}

Decimal form:

h = 4, - 0.667

h=4 , -0.667

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

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

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:

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

2. Solve the two equations for h

13 additional steps

$(4 h - 2) = 2 \cdot (h + 3)$

Expand the parentheses:

$(4 h - 2) = 2 h + 2 \cdot 3$

Simplify the arithmetic:

$(4 h - 2) = 2 h + 6$

Subtract from both sides:

$(4 h - 2) - 2 h = (2 h + 6) - 2 h$

Group like terms:

$(4 h - 2 h) - 2 = (2 h + 6) - 2 h$

Simplify the arithmetic:

$2 h - 2 = (2 h + 6) - 2 h$

Group like terms:

$2 h - 2 = (2 h - 2 h) + 6$

Simplify the arithmetic:

$2 h - 2 = 6$

Add to both sides:

$(2 h - 2) + 2 = 6 + 2$

Simplify the arithmetic:

$2 h = 6 + 2$

Simplify the arithmetic:

$2 h = 8$

Divide both sides by :

$\frac{(2 h)}{2} = \frac{8}{2}$

Simplify the fraction:

$h = \frac{8}{2}$

Find the greatest common factor of the numerator and denominator:

$h = \frac{(4 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$h = 4$

16 additional steps

$(4 h - 2) = 2 \cdot (- (h + 3))$

Expand the parentheses:

$(4 h - 2) = 2 \cdot (- h - 3)$

$(4 h - 2) = 2 \cdot - h + 2 \cdot - 3$

Group like terms:

$(4 h - 2) = (2 \cdot - 1) h + 2 \cdot - 3$

Multiply the coefficients:

$(4 h - 2) = - 2 h + 2 \cdot - 3$

Simplify the arithmetic:

$(4 h - 2) = - 2 h - 6$

Add to both sides:

$(4 h - 2) + 2 h = (- 2 h - 6) + 2 h$

Group like terms:

$(4 h + 2 h) - 2 = (- 2 h - 6) + 2 h$

Simplify the arithmetic:

$6 h - 2 = (- 2 h - 6) + 2 h$

Group like terms:

$6 h - 2 = (- 2 h + 2 h) - 6$

Simplify the arithmetic:

$6 h - 2 = - 6$

Add to both sides:

$(6 h - 2) + 2 = - 6 + 2$

Simplify the arithmetic:

$6 h = - 6 + 2$

Simplify the arithmetic:

$6 h = - 4$

Divide both sides by :

$\frac{(6 h)}{6} = \frac{- 4}{6}$

Simplify the fraction:

$h = \frac{- 4}{6}$

Find the greatest common factor of the numerator and denominator:

$h = \frac{(- 2 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

$h = \frac{- 2}{3}$

3. List the solutions

$h = 4, - \frac{2}{3}$
(2 solution(s))

4. Graph

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

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 h

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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