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Solution - Absolute value equations

Exact form:

a = 3, - \frac{11}{3}

a=3 , -\frac{11}{3}

Mixed number form:

a = 3, - 3 \frac{2}{3}

a=3 , -3\frac{2}{3}

Decimal form:

a = 3, - 3.667

a=3 , -3.667

See steps

Step-by-step explanation

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

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

Add $| a + 7 |$ to both sides of the equation:

$2 | a + 2 | - | a + 7 | + | a + 7 | = | a + 7 |$

Simplify the arithmetic

$2 | a + 2 | = | a + 7 |$

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

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

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:

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

3. Solve the two equations for a

9 additional steps

$2 \cdot (a + 2) = (a + 7)$

Expand the parentheses:

$2 a + 2 \cdot 2 = (a + 7)$

Simplify the arithmetic:

$2 a + 4 = (a + 7)$

Subtract from both sides:

$(2 a + 4) - a = (a + 7) - a$

Group like terms:

$(2 a - a) + 4 = (a + 7) - a$

Simplify the arithmetic:

$a + 4 = (a + 7) - a$

Group like terms:

$a + 4 = (a - a) + 7$

Simplify the arithmetic:

$a + 4 = 7$

Subtract from both sides:

$(a + 4) - 4 = 7 - 4$

Simplify the arithmetic:

$a = 7 - 4$

Simplify the arithmetic:

$a = 3$

12 additional steps

$2 \cdot (a + 2) = (- (a + 7))$

Expand the parentheses:

$2 a + 2 \cdot 2 = (- (a + 7))$

Simplify the arithmetic:

$2 a + 4 = (- (a + 7))$

Expand the parentheses:

$2 a + 4 = - a - 7$

Add to both sides:

$(2 a + 4) + a = (- a - 7) + a$

Group like terms:

$(2 a + a) + 4 = (- a - 7) + a$

Simplify the arithmetic:

$3 a + 4 = (- a - 7) + a$

Group like terms:

$3 a + 4 = (- a + a) - 7$

Simplify the arithmetic:

$3 a + 4 = - 7$

Subtract from both sides:

$(3 a + 4) - 4 = - 7 - 4$

Simplify the arithmetic:

$3 a = - 7 - 4$

Simplify the arithmetic:

$3 a = - 11$

Divide both sides by :

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

Simplify the fraction:

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

4. List the solutions

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

5. Graph

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

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 a

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved