Precalculus Examples

$\frac{1}{x - 2} - \frac{1}{x - 3} = - \frac{1}{2}$

Step 1

Find the LCD of the terms in the equation.

Tap for more steps...

Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x - 2, x - 3, 2$

Step 1.2

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 1.3

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 1.4

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

Step 1.5

The LCM of $1, 1, 2$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2$

Step 1.6

The factor for $x - 2$ is $x - 2$ itself.

$(x - 2) = x - 2$

$(x - 2)$ occurs $1$ time.

Step 1.7

The factor for $x - 3$ is $x - 3$ itself.

$(x - 3) = x - 3$

$(x - 3)$ occurs $1$ time.

Step 1.8

The LCM of $x - 2, x - 3$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(x - 2) (x - 3)$

Step 1.9

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$2 (x - 2) (x - 3)$

Step 2

Multiply each term in $\frac{1}{x - 2} - \frac{1}{x - 3} = - \frac{1}{2}$ by $2 (x - 2) (x - 3)$ to eliminate the fractions.

Tap for more steps...

Step 2.1

Multiply each term in $\frac{1}{x - 2} - \frac{1}{x - 3} = - \frac{1}{2}$ by $2 (x - 2) (x - 3)$ .

$\frac{1}{x - 2} (2 (x - 2) (x - 3)) - \frac{1}{x - 3} (2 (x - 2) (x - 3)) = - \frac{1}{2} (2 (x - 2) (x - 3))$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Rewrite using the commutative property of multiplication.

$2 \frac{1}{x - 2} ((x - 2) (x - 3)) - \frac{1}{x - 3} (2 (x - 2) (x - 3)) = - \frac{1}{2} (2 (x - 2) (x - 3))$

Step 2.2.1.2

Combine $2$ and $\frac{1}{x - 2}$ .

$\frac{2}{x - 2} ((x - 2) (x - 3)) - \frac{1}{x - 3} (2 (x - 2) (x - 3)) = - \frac{1}{2} (2 (x - 2) (x - 3))$

Step 2.2.1.3

Cancel the common factor of $x - 2$ .

Tap for more steps...

Step 2.2.1.3.1

Cancel the common factor.

$\frac{2}{x - 2} ((x - 2) (x - 3)) - \frac{1}{x - 3} (2 (x - 2) (x - 3)) = - \frac{1}{2} (2 (x - 2) (x - 3))$

Step 2.2.1.3.2

Rewrite the expression.

$2 (x - 3) - \frac{1}{x - 3} (2 (x - 2) (x - 3)) = - \frac{1}{2} (2 (x - 2) (x - 3))$

Step 2.2.1.4

Apply the distributive property.

$2 x + 2 \cdot - 3 - \frac{1}{x - 3} (2 (x - 2) (x - 3)) = - \frac{1}{2} (2 (x - 2) (x - 3))$

Step 2.2.1.5

Multiply $2$ by $- 3$ .

$2 x - 6 - \frac{1}{x - 3} (2 (x - 2) (x - 3)) = - \frac{1}{2} (2 (x - 2) (x - 3))$

Step 2.2.1.6

Cancel the common factor of $x - 3$ .

Tap for more steps...

Step 2.2.1.6.1

Move the leading negative in $- \frac{1}{x - 3}$ into the numerator.

$2 x - 6 + \frac{- 1}{x - 3} (2 (x - 2) (x - 3)) = - \frac{1}{2} (2 (x - 2) (x - 3))$

Step 2.2.1.6.2

Factor $x - 3$ out of $2 (x - 2) (x - 3)$ .

$2 x - 6 + \frac{- 1}{x - 3} ((x - 3) (2 (x - 2))) = - \frac{1}{2} (2 (x - 2) (x - 3))$

Step 2.2.1.6.3

Cancel the common factor.

$2 x - 6 + \frac{- 1}{x - 3} ((x - 3) (2 (x - 2))) = - \frac{1}{2} (2 (x - 2) (x - 3))$

Step 2.2.1.6.4

Rewrite the expression.

$2 x - 6 - (2 (x - 2)) = - \frac{1}{2} (2 (x - 2) (x - 3))$

Step 2.2.1.7

Multiply $2$ by $- 1$ .

$2 x - 6 - 2 (x - 2) = - \frac{1}{2} (2 (x - 2) (x - 3))$

Step 2.2.1.8

Apply the distributive property.

$2 x - 6 - 2 x - 2 \cdot - 2 = - \frac{1}{2} (2 (x - 2) (x - 3))$

Step 2.2.1.9

Multiply $- 2$ by $- 2$ .

$2 x - 6 - 2 x + 4 = - \frac{1}{2} (2 (x - 2) (x - 3))$

Step 2.2.2

Simplify by adding terms.

Tap for more steps...

Step 2.2.2.1

Combine the opposite terms in $2 x - 6 - 2 x + 4$ .

Tap for more steps...

Step 2.2.2.1.1

Subtract $2 x$ from $2 x$ .

$0 - 6 + 4 = - \frac{1}{2} (2 (x - 2) (x - 3))$

Step 2.2.2.1.2

Subtract $6$ from $0$ .

$- 6 + 4 = - \frac{1}{2} (2 (x - 2) (x - 3))$

Step 2.2.2.2

Add $- 6$ and $4$ .

$- 2 = - \frac{1}{2} (2 (x - 2) (x - 3))$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.1.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$- 2 = \frac{- 1}{2} (2 (x - 2) (x - 3))$

Step 2.3.1.2

Factor $2$ out of $2 (x - 2) (x - 3)$ .

$- 2 = \frac{- 1}{2} (2 ((x - 2) (x - 3)))$

Step 2.3.1.3

Cancel the common factor.

$- 2 = \frac{- 1}{2} (2 ((x - 2) (x - 3)))$

Step 2.3.1.4

Rewrite the expression.

$- 2 = - ((x - 2) (x - 3))$

Step 2.3.2

Expand $(x - 2) (x - 3)$ using the FOIL Method.

Tap for more steps...

Step 2.3.2.1

Apply the distributive property.

$- 2 = - (x (x - 3) - 2 (x - 3))$

Step 2.3.2.2

Apply the distributive property.

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

Step 2.3.2.3

Apply the distributive property.

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

Step 2.3.3

Simplify and combine like terms.

Tap for more steps...

Step 2.3.3.1

Simplify each term.

Tap for more steps...

Step 2.3.3.1.1

Multiply $x$ by $x$ .

$- 2 = - (x^{2} + x \cdot - 3 - 2 x - 2 \cdot - 3)$

Step 2.3.3.1.2

Move $- 3$ to the left of $x$ .

$- 2 = - (x^{2} - 3 \cdot x - 2 x - 2 \cdot - 3)$

Step 2.3.3.1.3

Multiply $- 2$ by $- 3$ .

$- 2 = - (x^{2} - 3 x - 2 x + 6)$

Step 2.3.3.2

Subtract $2 x$ from $- 3 x$ .

$- 2 = - (x^{2} - 5 x + 6)$

Step 2.3.4

Apply the distributive property.

$- 2 = - x^{2} - (- 5 x) - 1 \cdot 6$

Step 2.3.5

Simplify.

Tap for more steps...

Step 2.3.5.1

Multiply $- 5$ by $- 1$ .

$- 2 = - x^{2} + 5 x - 1 \cdot 6$

Step 2.3.5.2

Multiply $- 1$ by $6$ .

$- 2 = - x^{2} + 5 x - 6$

Step 3

Solve the equation.

Tap for more steps...

Step 3.1

Rewrite the equation as $- x^{2} + 5 x - 6 = - 2$ .

$- x^{2} + 5 x - 6 = - 2$

Step 3.2

Add $2$ to both sides of the equation.

$- x^{2} + 5 x - 6 + 2 = 0$

Step 3.3

Add $- 6$ and $2$ .

$- x^{2} + 5 x - 4 = 0$

Step 3.4

Factor the left side of the equation.

Tap for more steps...

Step 3.4.1

Factor $- 1$ out of $- x^{2} + 5 x - 4$ .

Tap for more steps...

Step 3.4.1.1

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + 5 x - 4 = 0$

Step 3.4.1.2

Factor $- 1$ out of $5 x$ .

$- (x^{2}) - (- 5 x) - 4 = 0$

Step 3.4.1.3

Rewrite $- 4$ as $- 1 (4)$ .

$- (x^{2}) - (- 5 x) - 1 \cdot 4 = 0$

Step 3.4.1.4

Factor $- 1$ out of $- (x^{2}) - (- 5 x)$ .

$- (x^{2} - 5 x) - 1 \cdot 4 = 0$

Step 3.4.1.5

Factor $- 1$ out of $- (x^{2} - 5 x) - 1 (4)$ .

$- (x^{2} - 5 x + 4) = 0$

Step 3.4.2

Factor.

Tap for more steps...

Step 3.4.2.1

Factor $x^{2} - 5 x + 4$ using the AC method.

Tap for more steps...

Step 3.4.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $4$ and whose sum is $- 5$ .

$- 4, - 1$

Step 3.4.2.1.2

Write the factored form using these integers.

$- ((x - 4) (x - 1)) = 0$

Step 3.4.2.2

Remove unnecessary parentheses.

$- (x - 4) (x - 1) = 0$

Step 3.5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 4 = 0$

$x - 1 = 0$

Step 3.6

Set $x - 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.6.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 3.6.2

Add $4$ to both sides of the equation.

$x = 4$

Step 3.7

Set $x - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.7.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 3.7.2

Add $1$ to both sides of the equation.

$x = 1$

Step 3.8

The final solution is all the values that make $- (x - 4) (x - 1) = 0$ true.

$x = 4, 1$