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Solution - Absolute value equations

Exact form:

x = 10, - \frac{4}{9}

x=10 , -\frac{4}{9}

Decimal form:

x = 10, - 0.444

x=10 , -0.444

See steps

Step-by-step explanation

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

$| 5 x - 3 | - | 4 x + 7 | = 0$

Add $| 4 x + 7 |$ to both sides of the equation:

$| 5 x - 3 | - | 4 x + 7 | + | 4 x + 7 | = | 4 x + 7 |$

Simplify the arithmetic

$| 5 x - 3 | = | 4 x + 7 |$

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
$| 5 x - 3 | = | 4 x + 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 x - 3 \| = \| 4 x + 7 \|$
$x = + y$	$(5 x - 3) = (4 x + 7)$
$x = - y$	$(5 x - 3) = (- (4 x + 7))$
$+ x = y$	$(5 x - 3) = (4 x + 7)$
$- x = y$	$- (5 x - 3) = (4 x + 7)$

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:

$\| x \| = \| y \|$	$\| 5 x - 3 \| = \| 4 x + 7 \|$
$x = + y$ , $+ x = y$	$(5 x - 3) = (4 x + 7)$
$x = - y$ , $- x = y$	$(5 x - 3) = (- (4 x + 7))$

3. Solve the two equations for x

7 additional steps

$(5 x - 3) = (4 x + 7)$

Subtract from both sides:

$(5 x - 3) - 4 x = (4 x + 7) - 4 x$

Group like terms:

$(5 x - 4 x) - 3 = (4 x + 7) - 4 x$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$x - 3 = 7$

Add to both sides:

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

Simplify the arithmetic:

$x = 7 + 3$

Simplify the arithmetic:

$x = 10$

10 additional steps

$(5 x - 3) = - (4 x + 7)$

Expand the parentheses:

$(5 x - 3) = - 4 x - 7$

Add to both sides:

$(5 x - 3) + 4 x = (- 4 x - 7) + 4 x$

Group like terms:

$(5 x + 4 x) - 3 = (- 4 x - 7) + 4 x$

Simplify the arithmetic:

$9 x - 3 = (- 4 x - 7) + 4 x$

Group like terms:

$9 x - 3 = (- 4 x + 4 x) - 7$

Simplify the arithmetic:

$9 x - 3 = - 7$

Add to both sides:

$(9 x - 3) + 3 = - 7 + 3$

Simplify the arithmetic:

$9 x = - 7 + 3$

Simplify the arithmetic:

$9 x = - 4$

Divide both sides by :

$\frac{(9 x)}{9} = \frac{- 4}{9}$

Simplify the fraction:

$x = \frac{- 4}{9}$

4. List the solutions

$x = 10, - \frac{4}{9}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 5 x - 3 |$
$y = | 4 x + 7 |$
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