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Solution - Absolute value equations

Exact form:

x = \frac{1}{18}, \frac{3}{4}

x=\frac{1}{18} , \frac{3}{4}

Decimal form:

x = 0.056, 0.75

x=0.056 , 0.75

See steps

Step-by-step explanation

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

$| 3 x + 4 | - 5 | - 3 x + 1 | = 0$

Add $5 | - 3 x + 1 |$ to both sides of the equation:

$| 3 x + 4 | - 5 | - 3 x + 1 | + 5 | - 3 x + 1 | = 5 | - 3 x + 1 |$

Simplify the arithmetic

$| 3 x + 4 | = 5 | - 3 x + 1 |$

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
$| 3 x + 4 | = 5 | - 3 x + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 x + 4 \| = 5 \| - 3 x + 1 \|$
$x = + y$	$(3 x + 4) = 5 (- 3 x + 1)$
$x = - y$	$(3 x + 4) = 5 (- (- 3 x + 1))$
$+ x = y$	$(3 x + 4) = 5 (- 3 x + 1)$
$- x = y$	$- (3 x + 4) = 5 (- 3 x + 1)$

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:

$\| x \| = \| y \|$	$\| 3 x + 4 \| = 5 \| - 3 x + 1 \|$
$x = + y$ , $+ x = y$	$(3 x + 4) = 5 (- 3 x + 1)$
$x = - y$ , $- x = y$	$(3 x + 4) = 5 (- (- 3 x + 1))$

3. Solve the two equations for x

12 additional steps

$(3 x + 4) = 5 \cdot (- 3 x + 1)$

Expand the parentheses:

$(3 x + 4) = 5 \cdot - 3 x + 5 \cdot 1$

Multiply the coefficients:

$(3 x + 4) = - 15 x + 5 \cdot 1$

Simplify the arithmetic:

$(3 x + 4) = - 15 x + 5$

Add to both sides:

$(3 x + 4) + 15 x = (- 15 x + 5) + 15 x$

Group like terms:

$(3 x + 15 x) + 4 = (- 15 x + 5) + 15 x$

Simplify the arithmetic:

$18 x + 4 = (- 15 x + 5) + 15 x$

Group like terms:

$18 x + 4 = (- 15 x + 15 x) + 5$

Simplify the arithmetic:

$18 x + 4 = 5$

Subtract from both sides:

$(18 x + 4) - 4 = 5 - 4$

Simplify the arithmetic:

$18 x = 5 - 4$

Simplify the arithmetic:

$18 x = 1$

Divide both sides by :

$\frac{(18 x)}{18} = \frac{1}{18}$

Simplify the fraction:

$x = \frac{1}{18}$

17 additional steps

$(3 x + 4) = 5 \cdot (- (- 3 x + 1))$

Expand the parentheses:

$(3 x + 4) = 5 \cdot (3 x - 1)$

Expand the parentheses:

$(3 x + 4) = 5 \cdot 3 x + 5 \cdot - 1$

Multiply the coefficients:

$(3 x + 4) = 15 x + 5 \cdot - 1$

Simplify the arithmetic:

$(3 x + 4) = 15 x - 5$

Subtract from both sides:

$(3 x + 4) - 15 x = (15 x - 5) - 15 x$

Group like terms:

$(3 x - 15 x) + 4 = (15 x - 5) - 15 x$

Simplify the arithmetic:

$- 12 x + 4 = (15 x - 5) - 15 x$

Group like terms:

$- 12 x + 4 = (15 x - 15 x) - 5$

Simplify the arithmetic:

$- 12 x + 4 = - 5$

Subtract from both sides:

$(- 12 x + 4) - 4 = - 5 - 4$

Simplify the arithmetic:

$- 12 x = - 5 - 4$

Simplify the arithmetic:

$- 12 x = - 9$

Divide both sides by :

$\frac{(- 12 x)}{- 12} = \frac{- 9}{- 12}$

Cancel out the negatives:

$\frac{12 x}{12} = \frac{- 9}{- 12}$

Simplify the fraction:

$x = \frac{- 9}{- 12}$

Cancel out the negatives:

$x = \frac{9}{12}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(3 \cdot 3)}{(4 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = \frac{3}{4}$

4. List the solutions

$x = \frac{1}{18}, \frac{3}{4}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 3 x + 4 |$
$y = 5 | - 3 x + 1 |$
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