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Solution - Absolute value equations

Exact form:

n = \frac{9}{2}

n=\frac{9}{2}

Mixed number form:

n = 4 \frac{1}{2}

n=4\frac{1}{2}

Decimal form:

n = 4.5

n=4.5

See steps

Step-by-step explanation

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

$| n - 7 | - | - n + 2 | = 0$

Add $| - n + 2 |$ to both sides of the equation:

$| n - 7 | - | - n + 2 | + | - n + 2 | = | - n + 2 |$

Simplify the arithmetic

$| n - 7 | = | - n + 2 |$

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
$| n - 7 | = | - n + 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| n - 7 \| = \| - n + 2 \|$
$x = + y$	$(n - 7) = (- n + 2)$
$x = - y$	$(n - 7) = (- (- n + 2))$
$+ x = y$	$(n - 7) = (- n + 2)$
$- x = y$	$- (n - 7) = (- n + 2)$

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:

$\| x \| = \| y \|$	$\| n - 7 \| = \| - n + 2 \|$
$x = + y$ , $+ x = y$	$(n - 7) = (- n + 2)$
$x = - y$ , $- x = y$	$(n - 7) = (- (- n + 2))$

3. Solve the two equations for n

9 additional steps

$(n - 7) = (- n + 2)$

Add to both sides:

$(n - 7) + n = (- n + 2) + n$

Group like terms:

$(n + n) - 7 = (- n + 2) + n$

Simplify the arithmetic:

$2 n - 7 = (- n + 2) + n$

Group like terms:

$2 n - 7 = (- n + n) + 2$

Simplify the arithmetic:

$2 n - 7 = 2$

Add to both sides:

$(2 n - 7) + 7 = 2 + 7$

Simplify the arithmetic:

$2 n = 2 + 7$

Simplify the arithmetic:

$2 n = 9$

Divide both sides by :

$\frac{(2 n)}{2} = \frac{9}{2}$

Simplify the fraction:

$n = \frac{9}{2}$

6 additional steps

$(n - 7) = - (- n + 2)$

Expand the parentheses:

$(n - 7) = n - 2$

Subtract from both sides:

$(n - 7) - n = (n - 2) - n$

Group like terms:

$(n - n) - 7 = (n - 2) - n$

Simplify the arithmetic:

$- 7 = (n - 2) - n$

Group like terms:

$- 7 = (n - n) - 2$

Simplify the arithmetic:

$- 7 = - 2$

The statement is false:

$- 7 = - 2$

The equation is false so it has no solution.

4. List the solutions

$n = \frac{9}{2}$
(1 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | n - 7 |$
$y = | - n + 2 |$
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 n

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 n

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved