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Solution - Absolute value equations

Exact form:

h = - 1, 4

h=-1 , 4

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| 4 h + 9 | + | 6 h + 1 | = 0$

Add $- | 6 h + 1 |$ to both sides of the equation:

$| 4 h + 9 | + | 6 h + 1 | - | 6 h + 1 | = - | 6 h + 1 |$

Simplify the arithmetic

$| 4 h + 9 | = - | 6 h + 1 |$

2. 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
$| 4 h + 9 | = - | 6 h + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 h + 9 \| = - \| 6 h + 1 \|$
$x = + y$	$(4 h + 9) = - (6 h + 1)$
$x = - y$	$(4 h + 9) = - - (6 h + 1)$
$+ x = y$	$(4 h + 9) = - (6 h + 1)$
$- x = y$	$- (4 h + 9) = - (6 h + 1)$

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:

$\| x \| = \| y \|$	$\| 4 h + 9 \| = - \| 6 h + 1 \|$
$x = + y$ , $+ x = y$	$(4 h + 9) = - (6 h + 1)$
$x = - y$ , $- x = y$	$(4 h + 9) = - - (6 h + 1)$

3. Solve the two equations for h

11 additional steps

$(4 h + 9) = - (6 h + 1)$

Expand the parentheses:

$(4 h + 9) = - 6 h - 1$

Add to both sides:

$(4 h + 9) + 6 h = (- 6 h - 1) + 6 h$

Group like terms:

$(4 h + 6 h) + 9 = (- 6 h - 1) + 6 h$

Simplify the arithmetic:

$10 h + 9 = (- 6 h - 1) + 6 h$

Group like terms:

$10 h + 9 = (- 6 h + 6 h) - 1$

Simplify the arithmetic:

$10 h + 9 = - 1$

Subtract from both sides:

$(10 h + 9) - 9 = - 1 - 9$

Simplify the arithmetic:

$10 h = - 1 - 9$

Simplify the arithmetic:

$10 h = - 10$

Divide both sides by :

$\frac{(10 h)}{10} = \frac{- 10}{10}$

Simplify the fraction:

$h = \frac{- 10}{10}$

Simplify the fraction:

$h = - 1$

14 additional steps

$(4 h + 9) = - (- (6 h + 1))$

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

$(4 h + 9) = 6 h + 1$

Subtract from both sides:

$(4 h + 9) - 6 h = (6 h + 1) - 6 h$

Group like terms:

$(4 h - 6 h) + 9 = (6 h + 1) - 6 h$

Simplify the arithmetic:

$- 2 h + 9 = (6 h + 1) - 6 h$

Group like terms:

$- 2 h + 9 = (6 h - 6 h) + 1$

Simplify the arithmetic:

$- 2 h + 9 = 1$

Subtract from both sides:

$(- 2 h + 9) - 9 = 1 - 9$

Simplify the arithmetic:

$- 2 h = 1 - 9$

Simplify the arithmetic:

$- 2 h = - 8$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Cancel out the negatives:

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

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:

$h = 4$

4. List the solutions

$h = - 1, 4$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 4 h + 9 |$
$y = - | 6 h + 1 |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for h

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for h

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved