Problem 54 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The value of the expression is -21.

Step 1: Substitute the value of x

Replace the variable \( x \) in the expression with the given value \( x = -\frac{1}{2} \). The expression becomes:\[-3\left(-\frac{1}{2}+1\right)+4\left(\frac{1}{2}-2\right)-3\left(\frac{1}{2}+4\right)\]

Step 2: Simplify the expressions inside the parentheses

Calculate the value inside each set of parentheses:1. \(-\frac{1}{2} + 1 = \frac{1}{2}\)2. \(\frac{1}{2} - 2 = -\frac{3}{2}\)3. \(\frac{1}{2} + 4 = \frac{9}{2}\)Thus, the expression now is:\[-3\left(\frac{1}{2}\right) + 4\left(-\frac{3}{2}\right) - 3\left(\frac{9}{2}\right)\]

Step 3: Perform the multiplication

Multiply the numbers outside the parentheses with those inside:1. \(-3 \times \frac{1}{2} = -\frac{3}{2}\)2. \(4 \times -\frac{3}{2} = -6\)3. \(-3 \times \frac{9}{2} = -\frac{27}{2}\)The expression is now:\[-\frac{3}{2} - 6 - \frac{27}{2}\]

Step 4: Add/Subtract terms

Combine all terms together. First, convert \(-6\) into a fraction with a denominator of 2 for easier calculation:- \(6 = \frac{12}{2}\)Now add/subtract:\[-\frac{3}{2} - \frac{12}{2} - \frac{27}{2} = \frac{-3 - 12 - 27}{2} = \frac{-42}{2}\]Thus, the final result is \(-21\).

Key Concepts

Evaluating Expressions

Evaluating algebraic expressions is a common task in mathematics where you find the value of an expression by replacing variables with their given numbers. Consider an expression like \(-3(x+1)+4(-x-2)-3(-x+4)\). Here, you have an algebraic expression with a variable \(x\). To evaluate it:

• Identify the values of variables given in the problem. For our exercise, \(x = -\frac{1}{2}\).
• Substitute this value into the expression in place of \(x\).
• Solve the expression by performing the arithmetic operations according to the order of operations: parentheses, exponents (if any), multiplication and division, addition, and subtraction, often abbreviated as PEMDAS.

Understanding these steps makes it easier to approach any algebraic expression. It's much like a cooking recipe, follow the steps and you get the expected outcome.

Substituting Variables

Substituting variables is a fundamental skill in algebra that involves replacing a variable with a specific value. Doing this accurately ensures that the rest of the problem can be solved without error. In the example\(-3(x+1)+4(-x-2)-3(-x+4)\) with \(x=-\frac{1}{2}\):

• Simply insert \(-\frac{1}{2}\) into all instances of \(x\) within the expression.
• This transforms the expression to\[-3\left(-\frac{1}{2}+1\right) + 4\left(\frac{1}{2}-2\right) - 3\left(\frac{1}{2}+4\right)\]
• The new expression is free of variables and ready for simplification.

Remember to replace variables systematically and check your substitution, as a mistake here will lead to an incorrect final answer. This step reduces complexity and allows focus solely on computation.

Simplifying Expressions

Simplifying expressions is about reducing them to their simplest form, making calculation easier and clearer. Once the variable has been substituted, like in our problem where the expression post-substitution becomes:

• Compute the expressions inside the parentheses first: \(-\frac{1}{2}+1\) becomes \(\frac{1}{2}\), \(\frac{1}{2}-2\) becomes \(-\frac{3}{2}\), and \(\frac{1}{2}+4\) becomes \(\frac{9}{2}\).
• Next, perform the multiplications outside the parentheses. For instance, \(-3 \times \frac{1}{2} = -\frac{3}{2}\).
• Lastly, add or subtract the terms together using a common denominator, as seen when combining \(-\frac{3}{2} - 6 - \frac{27}{2}\).

Simplifying reveals the expression’s value, which in this case is \(-21\). Effectively simplifying expressions is key to problem-solving and helps in understanding the calculation.”