Problem 43 - Elementary Algebra

Question

For the following problems, use the zero-factor property to solve the equations. $$ -2(m+11)=0 $$

Step-by-Step Solution

Answer

Answer: The solution for the equation -2(m+11)=0 is m = -11.

Step 1: Identify the factors

First, let's identify the two factors in the given equation: $$ -2(m+11) = 0 $$ Here, the factors are -2 and (m+11).

Step 2: Apply the zero-factor property

Since the product of these two factors is zero, we will set each factor equal to zero: $$ -2 = 0 \quad \text{or} \quad m+11 = 0 $$

Step 3: Solve the equation for m

The first equation, -2=0, is not true, so it does not give us a valid solution for m. Now, let's solve the second equation for m: $$ m+11 = 0 $$ Subtracting 11 from both sides, we get: $$ m = -11 $$

Step 4: State the solution

The solution for the equation -2(m+11)=0 is m = -11.

Key Concepts

Equation Solving

Equation solving often involves finding the value (or values) of a variable that makes a mathematical statement true. In the case of this exercise, we're dealing with a linear equation. Solving involves isolating the variable to find its precise value.

• In our specific problem, we started with the equation \(-2(m+11)=0\).
• The goal was to find the value of \(m\) that makes this equation true.
• The Zero-Factor Property came in handy to simplify the approach.

Let's break it down a bit. Solving an equation by isolating the variable means manipulating the equation until you have the variable alone on one side of the equation sign. It usually involves performing inverse operations such as addition or division to cancel out other numbers or functions that are with the variable.

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In our exercise, algebra helps us express and solve the equation using letters and numbers.

• The equation \(-2(m+11)=0\) is an example of an algebraic expression where \(m\) is our variable.
• Algebra involves understanding the relationship between variables and constants, like numbers we add or subtract.

In the given problem, we use algebraic methods to distribute and simplify expressions. This allows us to set the framework for determining correct values, like setting \(m+11=0\) to find the solution. This process of using algebra makes it possible to find unknown numbers efficiently.

Factoring Equations

Factoring equations involves breaking down expressions into products of simpler expressions. When dealing with the zero-factor property, the idea is to express a polynomial as a product of its factors.

• With the equation \(-2(m+11)=0\), the property suggests that if a product of two factors is zero, then at least one of the factors must be zero.
• This means we can set each individual factor equal to zero and solve them separately.

For example, once we identified our factors \(-2\) and \(m+11\), we knew either of them could potentially make the entire expression equal zero. However, in practical terms, \(-2\) couldn't be zero, so we focused on \(m+11\). Solving the resulting simpler equation \(m+11=0\) gives us a straightforward path to find \(m = -11\). This strategy greatly simplifies the process of finding solutions in algebraic equations.