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Chapter 3: Problem 95

Solve the equation and check your solution. $$4(x+3)=0$$

Short Answer

Expert verified
The solution to the equation is \(x = -3\).

Step by step solution

01

Distribute the Multiplication

First, distribute the multiplication of 4 to the items within the parentheses, which gives us \(4x+12=0\).
02

Isolate the Variable

The next step is to isolate the variable 'x' on one side of the equation. To do so, subtract 12 from both sides of the equation. This gives us \(4x = -12\). Finally, divide both sides of the equation by 4 to isolate 'x'. This gives us \(x=-3\).
03

Check the Solution

To check if the solution is correct, substitute the value of 'x' into the original equation. So we get \(4(-3+3) = 4(0) = 0\), which confirms the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
Understanding the distributive property is essential when solving equations that involve parentheses. The distributive property helps us multiply a single term across terms inside a parenthesis. In the equation \(4(x+3)=0\), we need to distribute 4 across \(x+3\).
  • Start by multiplying 4 by \(x\), which gives you \(4x\).
  • Then, multiply 4 by 3, resulting in 12.
Putting these together gives you the equation \(4x + 12 = 0\). The distributive property is a handy tool to simplify expressions and is often your first step in equation solving when you see terms in parentheses. It's like unpacking a box of chocolates; you have to open up each section to see what's inside.
Isolate Variable
When solving equations, isolating the variable is crucial. After distributing, our goal is to get \(x\) all by itself. In the equation \(4x + 12 = 0\), start by removing the addition or subtraction around \(x\).
Subtract 12 from both sides to cancel out the +12, resulting in \(4x = -12\). Remember, whatever you do on one side, do on the other. After this, the equation simplifies to making it all about \(x\) by dividing both sides by 4. This gives \(x = -3\). You now have isolated \(x\), making it the subject of the equation. This step ensures that you're inching closer to solving the equation, like peeling the layers of an onion until you get to the core.
Checking Solutions
Checking your solution is a vital part of solving equations. It confirms that you haven't made any arithmetic mistakes along the way. For the equation \(4(x+3)=0\), substitute \(x = -3\) back into the original equation to verify your result.
Here's how: Plug \(-3\) into \(x+3\), giving \(-3+3=0\). Multiply by 4 and get \(4 \times 0 = 0\). Since the left side equals the right side (0 = 0), your solution is confirmed correct.
This step creates confidence in your answer and helps in identifying potential errors. It's like taking a victory lap after finishing a race, ensuring that you've crossed every checkpoint.

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