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Solution - Absolute value equations

Exact form:

b = - \frac{11}{3}, - 13

b=-\frac{11}{3} , -13

Mixed number form:

b = - 3 \frac{2}{3}, - 13

b=-3\frac{2}{3} , -13

Decimal form:

b = - 3.667, - 13

b=-3.667 , -13

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
$| 2 b + 12 | = - | b - 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 b + 12 \| = - \| b - 1 \|$
$x = + y$	$(2 b + 12) = - (b - 1)$
$x = - y$	$(2 b + 12) = - (- (b - 1))$
$+ x = y$	$(2 b + 12) = - (b - 1)$
$- x = y$	$- (2 b + 12) = - (b - 1)$

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:

$\| x \| = \| y \|$	$\| 2 b + 12 \| = - \| b - 1 \|$
$x = + y$ , $+ x = y$	$(2 b + 12) = - (b - 1)$
$x = - y$ , $- x = y$	$(2 b + 12) = - (- (b - 1))$

2. Solve the two equations for b

10 additional steps

$(2 b + 12) = - (b - 1)$

Expand the parentheses:

$(2 b + 12) = - b + 1$

Add to both sides:

$(2 b + 12) + b = (- b + 1) + b$

Group like terms:

$(2 b + b) + 12 = (- b + 1) + b$

Simplify the arithmetic:

$3 b + 12 = (- b + 1) + b$

Group like terms:

$3 b + 12 = (- b + b) + 1$

Simplify the arithmetic:

$3 b + 12 = 1$

Subtract from both sides:

$(3 b + 12) - 12 = 1 - 12$

Simplify the arithmetic:

$3 b = 1 - 12$

Simplify the arithmetic:

$3 b = - 11$

Divide both sides by :

$\frac{(3 b)}{3} = \frac{- 11}{3}$

Simplify the fraction:

$b = \frac{- 11}{3}$

8 additional steps

$(2 b + 12) = - (- (b - 1))$

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

$(2 b + 12) = b - 1$

Subtract from both sides:

$(2 b + 12) - b = (b - 1) - b$

Group like terms:

$(2 b - b) + 12 = (b - 1) - b$

Simplify the arithmetic:

$b + 12 = (b - 1) - b$

Group like terms:

$b + 12 = (b - b) - 1$

Simplify the arithmetic:

$b + 12 = - 1$

Subtract from both sides:

$(b + 12) - 12 = - 1 - 12$

Simplify the arithmetic:

$b = - 1 - 12$

Simplify the arithmetic:

$b = - 13$

3. List the solutions

$b = - \frac{11}{3}, - 13$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 b + 12 |$
$y = - | b - 1 |$
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 without absolute value bars

2. Solve the two equations for b

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 b

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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