Question

Solve each equation, if possible. $$ \frac{-4}{x+4}=\frac{-3}{x+6} $$

Short answer

x = -12

Step-by-step solution

- Set the cross products equal

To solve the equation \(\frac{-4}{x+4} = \frac{-3}{x+6}\), use the property of proportions which states that the cross products are equal. This gives us the equation: \[-4(x+6) = -3(x+4)\]

- Distribute the constants

Next, distribute the constants -4 and -3 to the terms inside the parentheses: \[-4x - 24 = -3x - 12\]

- Isolate the variable term

To isolate the variable term, first add 4x to both sides of the equation: \[-24 = x - 12\]

- Solve for x

Now, add 12 to both sides to solve for x: \[-24 + 12 = x\]\[-12 = x\]

Key concepts

cross multiplication

Cross multiplication is a handy technique to solve rational equations where you have a fraction on each side of the equality sign. Essentially, you are 'crossing' the terms to create a simpler equation. For the equation given, \(\frac{-4}{x+4} = \frac{-3}{x+6}\), cross multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction. This means we multiply \-4\ by \(x+6\) and \-3\ by \(x+4\), giving us the new equation: \-4(x+6) = -3(x+4)\. This step eliminates the denominators and allows us to work with a simpler algebraic equation.
Always remember:

• Cross multiply only when you have one fraction on each side.
• Multiply the numerator of one fraction by the denominator of the other.
• This step helps in removing the fractions, making it easier to isolate and solve for the variable.

distributing constants

Once you have performed cross multiplication, the next step usually involves distributing any constants present outside of the parentheses. Distributing constants means you multiply the constant by each term inside the parentheses. For our equation after cross multiplication: \-4(x+6) = -3(x+4)\, you distribute the constants -4 and -3:

• Distributing -4 across \(x + 6\) gives us \-4x - 24\.
• Distributing -3 across \(x + 4\) gives us \-3x - 12\.

Thus, the equation \-4(x+6) = -3(x+4)\ simplifies to \-4x - 24 = -3x - 12\.
Key points to remember:

• Ensure you multiply each term inside the parentheses by the constant.
• This helps in simplifying the equation further.

isolating variable

Once the constants have been distributed, the next step is to isolate the variable. Isolating the variable involves moving terms containing the variable to one side of the equation and constant terms to the other side. For our simplified equation \-4x - 24 = -3x - 12\, we want to isolate \(x\):

• Add \4x\ to both sides to get rid of the \-4x\ term on the left: \-24 = x - 12\.

Now, the variable is isolated on one side of the equation. Next, we can solve for \(x\) by performing arithmetic operations.
Key points to remember:

• Always perform the same operation on both sides of the equation to maintain equality.
• Focus on getting the variable term on one side of the equation.

finding common denominators

Though not explicitly needed in this example, finding common denominators is an essential skill in solving rational equations. It's particularly useful when the equation involves multiple fractions with different denominators. The goal is to find a common denominator so the fractions can be combined or simplified. Here’s how you do it:

• Identify the denominators of all fractions involved.
• Determine the least common denominator (LCD) of these fractions.
• Rewrite each fraction with the LCD as the new denominator by multiplying the numerator and the denominator by the necessary factors.

This method allows you to combine the fractions into a simpler form, making it easier to solve for the variable. Although not used directly in the given exercise, understanding this concept will be very beneficial for more complex problems involving rational equations.