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Solution - Absolute value equations

Exact form:

x = - \frac{9}{2}, - 1

x=-\frac{9}{2} , -1

Mixed number form:

x = - 4 \frac{1}{2}, - 1

x=-4\frac{1}{2} , -1

Decimal form:

x = - 4.5, - 1

x=-4.5 , -1

See steps

Step-by-step explanation

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

$| 3 x - 4 | - 7 | x + 2 | = 0$

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

$| 3 x - 4 | - 7 | x + 2 | + 7 | x + 2 | = 7 | x + 2 |$

Simplify the arithmetic

$| 3 x - 4 | = 7 | x + 2 |$

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
$| 3 x - 4 | = 7 | x + 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 x - 4 \| = 7 \| x + 2 \|$
$x = + y$	$(3 x - 4) = 7 (x + 2)$
$x = - y$	$(3 x - 4) = 7 (- (x + 2))$
$+ x = y$	$(3 x - 4) = 7 (x + 2)$
$- x = y$	$- (3 x - 4) = 7 (x + 2)$

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:

$\| x \| = \| y \|$	$\| 3 x - 4 \| = 7 \| x + 2 \|$
$x = + y$ , $+ x = y$	$(3 x - 4) = 7 (x + 2)$
$x = - y$ , $- x = y$	$(3 x - 4) = 7 (- (x + 2))$

3. Solve the two equations for x

15 additional steps

$(3 x - 4) = 7 \cdot (x + 2)$

Expand the parentheses:

$(3 x - 4) = 7 x + 7 \cdot 2$

Simplify the arithmetic:

$(3 x - 4) = 7 x + 14$

Subtract from both sides:

$(3 x - 4) - 7 x = (7 x + 14) - 7 x$

Group like terms:

$(3 x - 7 x) - 4 = (7 x + 14) - 7 x$

Simplify the arithmetic:

$- 4 x - 4 = (7 x + 14) - 7 x$

Group like terms:

$- 4 x - 4 = (7 x - 7 x) + 14$

Simplify the arithmetic:

$- 4 x - 4 = 14$

Add to both sides:

$(- 4 x - 4) + 4 = 14 + 4$

Simplify the arithmetic:

$- 4 x = 14 + 4$

Simplify the arithmetic:

$- 4 x = 18$

Divide both sides by :

$\frac{(- 4 x)}{- 4} = \frac{18}{- 4}$

Cancel out the negatives:

$\frac{4 x}{4} = \frac{18}{- 4}$

Simplify the fraction:

$x = \frac{18}{- 4}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 18}{4}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 9 \cdot 2)}{(2 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 9}{2}$

15 additional steps

$(3 x - 4) = 7 \cdot (- (x + 2))$

Expand the parentheses:

$(3 x - 4) = 7 \cdot (- x - 2)$

$(3 x - 4) = 7 \cdot - x + 7 \cdot - 2$

Group like terms:

$(3 x - 4) = (7 \cdot - 1) x + 7 \cdot - 2$

Multiply the coefficients:

$(3 x - 4) = - 7 x + 7 \cdot - 2$

Simplify the arithmetic:

$(3 x - 4) = - 7 x - 14$

Add to both sides:

$(3 x - 4) + 7 x = (- 7 x - 14) + 7 x$

Group like terms:

$(3 x + 7 x) - 4 = (- 7 x - 14) + 7 x$

Simplify the arithmetic:

$10 x - 4 = (- 7 x - 14) + 7 x$

Group like terms:

$10 x - 4 = (- 7 x + 7 x) - 14$

Simplify the arithmetic:

$10 x - 4 = - 14$

Add to both sides:

$(10 x - 4) + 4 = - 14 + 4$

Simplify the arithmetic:

$10 x = - 14 + 4$

Simplify the arithmetic:

$10 x = - 10$

Divide both sides by :

$\frac{(10 x)}{10} = \frac{- 10}{10}$

Simplify the fraction:

$x = \frac{- 10}{10}$

Simplify the fraction:

$x = - 1$

4. List the solutions

$x = - \frac{9}{2}, - 1$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 3 x - 4 |$
$y = 7 | x + 2 |$
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 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