Problem 51 - Intermediate Algebra

Question

Evaluate the algebraic expressions for the given values of the variables. $$ -2(x+4)-(2 x-1), \quad x=-3 $$

Step-by-Step Solution

Answer

The evaluated expression is 5.

Step 1: Substitute the Value of x

First, let's substitute the given value of \( x = -3 \) into the expression. Replace every \( x \) with \( -3 \) in the expression:\[ -2(-3+4)-(2(-3)-1) \]

Step 2: Simplify the Expressions Inside the Parentheses

Now we need to simplify the expressions inside the parentheses. For \( -3 + 4 \):\[ -3 + 4 = 1 \]For \( 2(-3) - 1 \):\[ 2(-3) = -6 \]\[ -6 - 1 = -7 \]So the expression now is:\[ -2(1) - (-7) \]

Step 3: Apply the Distributive Property

Apply the distributive property by multiplying the constants outside the parentheses with the results obtained inside.Calculate \( -2 \times 1 \):\[ -2 imes 1 = -2 \]The expression is now:\[ -2 - (-7) \]

Step 4: Simplify the Expression

Simplify the expression by resolving the double negative and performing the subtraction:The subtraction of a negative value equals addition:\[ -2 - (-7) = -2 + 7 \]Finally, calculate the result:\[ -2 + 7 = 5 \]

Key Concepts

Substitution Method

The substitution method is a useful technique in algebra when you need to evaluate expressions for specific variable values. It's like solving a puzzle by replacing a variable with its numerical value. For example, in our expression, we substituted the value of \( x = -3 \). To do this, simply take every instance of \( x \) in the expression and replace it with \(-3\). After substitution, the expression goes from \(-2(x+4)-(2x-1)\) to \(-2(-3+4)-(2(-3)-1)\). By using this method, we transform the original algebraic expression into a simpler arithmetic expression that is much easier to calculate. Breaking this down to each step helps to avoid mistakes and makes sure the computation is accurate.

Distributive Property

The distributive property is a key concept in algebra that allows you to simplify expressions by distributing a factor over other terms inside parentheses. It states that \( a(b+c) = ab + ac \). This becomes incredibly useful when dealing with expressions like \(-2(x+4)-(2x-1)\).

Here's how we apply it:

• First, look at the structure in parentheses. After substitution and simplification, we had \(-2(1) - (-7)\).
• Next, use the distributive property to multiply the number outside, \(-2\), by the term inside: \(-2 \times 1\) results in \(-2\).
• Remember, distributing a negative sign flips the sign of each product. This helped simplify to finally get \(-2 - (-7)\), which indicates we subtract a negative, leading us to addition.

This property helps in breaking down complex expressions into manageable pieces, making calculations easier.

Simplifying Expressions

Simplifying expressions is an essential step in solving mathematical problems. It means making the expression as straightforward as possible without changing its value. Once you substitute the variable values and distribute any factors, the next part is simplifying any remaining terms.

Let's consider the expression \(-2 - (-7)\). The presence of two negative signs can be tricky. Remember, subtracting a negative is the same as adding the opposite. This is because of the rule: minus a negative equals a positive. Therefore, \(-2 - (-7)\) simplifies to \(-2 + 7\).

Here's how to approach simplifying:

• First, resolve any double negatives or combined terms.
• Next, perform the simple arithmetic to get to the answer. So, in this case, \(-2 + 7\) equals \(5\).

Through simplification, you conclude the problem-solving process with a clear and easy-to-understand result.