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Solution - Absolute value equations

Exact form:

x = - 29, \frac{1}{11}

x=-29 , \frac{1}{11}

Decimal form:

x = - 29, 0.091

x=-29 , 0.091

See steps

Step-by-step explanation

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

$5 | x - 3 | - 2 | 3 x + 7 | = 0$

Add $2 | 3 x + 7 |$ to both sides of the equation:

$5 | x - 3 | - 2 | 3 x + 7 | + 2 | 3 x + 7 | = 2 | 3 x + 7 |$

Simplify the arithmetic

$5 | x - 3 | = 2 | 3 x + 7 |$

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
$5 | x - 3 | = 2 | 3 x + 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$5 \| x - 3 \| = 2 \| 3 x + 7 \|$
$x = + y$	$5 (x - 3) = 2 (3 x + 7)$
$x = - y$	$5 (x - 3) = 2 (- (3 x + 7))$
$+ x = y$	$5 (x - 3) = 2 (3 x + 7)$
$- x = y$	$5 (- (x - 3)) = 2 (3 x + 7)$

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:

$\| x \| = \| y \|$	$5 \| x - 3 \| = 2 \| 3 x + 7 \|$
$x = + y$ , $+ x = y$	$5 (x - 3) = 2 (3 x + 7)$
$x = - y$ , $- x = y$	$5 (x - 3) = 2 (- (3 x + 7))$

3. Solve the two equations for x

15 additional steps

$5 \cdot (x - 3) = 2 \cdot (3 x + 7)$

Expand the parentheses:

$5 x + 5 \cdot - 3 = 2 \cdot (3 x + 7)$

Simplify the arithmetic:

$5 x - 15 = 2 \cdot (3 x + 7)$

Expand the parentheses:

$5 x - 15 = 2 \cdot 3 x + 2 \cdot 7$

Multiply the coefficients:

$5 x - 15 = 6 x + 2 \cdot 7$

Simplify the arithmetic:

$5 x - 15 = 6 x + 14$

Subtract from both sides:

$(5 x - 15) - 6 x = (6 x + 14) - 6 x$

Group like terms:

$(5 x - 6 x) - 15 = (6 x + 14) - 6 x$

Simplify the arithmetic:

$- x - 15 = (6 x + 14) - 6 x$

Group like terms:

$- x - 15 = (6 x - 6 x) + 14$

Simplify the arithmetic:

$- x - 15 = 14$

Add to both sides:

$(- x - 15) + 15 = 14 + 15$

Simplify the arithmetic:

$- x = 14 + 15$

Simplify the arithmetic:

$- x = 29$

Multiply both sides by :

$- x \cdot - 1 = 29 \cdot - 1$

Remove the one(s):

$x = 29 \cdot - 1$

Simplify the arithmetic:

$x = - 29$

15 additional steps

$5 \cdot (x - 3) = 2 \cdot (- (3 x + 7))$

Expand the parentheses:

$5 x + 5 \cdot - 3 = 2 \cdot (- (3 x + 7))$

Simplify the arithmetic:

$5 x - 15 = 2 \cdot (- (3 x + 7))$

Expand the parentheses:

$5 x - 15 = 2 \cdot (- 3 x - 7)$

Expand the parentheses:

$5 x - 15 = 2 \cdot - 3 x + 2 \cdot - 7$

Multiply the coefficients:

$5 x - 15 = - 6 x + 2 \cdot - 7$

Simplify the arithmetic:

$5 x - 15 = - 6 x - 14$

Add to both sides:

$(5 x - 15) + 6 x = (- 6 x - 14) + 6 x$

Group like terms:

$(5 x + 6 x) - 15 = (- 6 x - 14) + 6 x$

Simplify the arithmetic:

$11 x - 15 = (- 6 x - 14) + 6 x$

Group like terms:

$11 x - 15 = (- 6 x + 6 x) - 14$

Simplify the arithmetic:

$11 x - 15 = - 14$

Add to both sides:

$(11 x - 15) + 15 = - 14 + 15$

Simplify the arithmetic:

$11 x = - 14 + 15$

Simplify the arithmetic:

$11 x = 1$

Divide both sides by :

$\frac{(11 x)}{11} = \frac{1}{11}$

Simplify the fraction:

$x = \frac{1}{11}$

4. List the solutions

$x = - 29, \frac{1}{11}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = 5 | x - 3 |$
$y = 2 | 3 x + 7 |$
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 x

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 x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved