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Solution - Absolute value equations

Exact form:

b = \frac{7}{2}

b=\frac{7}{2}

Mixed number form:

b = 3 \frac{1}{2}

b=3\frac{1}{2}

Decimal form:

b = 3.5

b=3.5

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
$| b - 7 | = | b |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| b - 7 \| = \| b \|$
$x = + y$	$(b - 7) = (b)$
$x = - y$	$(b - 7) = - (b)$
$+ x = y$	$(b - 7) = (b)$
$- x = y$	$- (b - 7) = (b)$

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:

$\| x \| = \| y \|$	$\| b - 7 \| = \| b \|$
$x = + y$ , $+ x = y$	$(b - 7) = (b)$
$x = - y$ , $- x = y$	$(b - 7) = - (b)$

2. Solve the two equations for b

4 additional steps

$(b - 7) = b$

Subtract from both sides:

$(b - 7) - b = b - b$

Group like terms:

$(b - b) - 7 = b - b$

Simplify the arithmetic:

$- 7 = b - b$

Simplify the arithmetic:

$- 7 = 0$

The statement is false:

$- 7 = 0$

The equation is false so it has no solution.

8 additional steps

$(b - 7) = - b$

Add to both sides:

$(b - 7) + b = - b + b$

Group like terms:

$(b + b) - 7 = - b + b$

Simplify the arithmetic:

$2 b - 7 = - b + b$

Simplify the arithmetic:

$2 b - 7 = 0$

Add to both sides:

$(2 b - 7) + 7 = 0 + 7$

Simplify the arithmetic:

$2 b = 0 + 7$

Simplify the arithmetic:

$2 b = 7$

Divide both sides by :

$\frac{(2 b)}{2} = \frac{7}{2}$

Simplify the fraction:

$b = \frac{7}{2}$

3. Graph

Each line represents the function of one side of the equation:
$y = | b - 7 |$
$y = | b |$
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. 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. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved