Problem 11 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression for each value of \(x\). (If not possible, state the reason.)\(\begin{array}{lll}4 x-6 & \text { (a) } x=-1 & \text { (b) } x=0\end{array}\)

Step-by-Step Solution

Answer

For \(x = -1\), the value of the expression is -10. For \(x = 0\), the value is -6.

Step 1: Substitute \(x = -1\)

Replace \(x\) in \(4x - 6\) with -1. This gives \(4(-1) - 6\), which simplifies as -4 - 6.

Step 2: Compute for \(x = -1\)

Compute -4 - 6, which equals -10.

Step 3: Substitute \(x = 0\)

Now let's handle the second part of the problem. Replace \(x\) in \(4x - 6\) with 0, which gives \(4(0) - 6\). This simplifies as 0 - 6.

Step 4: Compute for \(x = 0\)

Compute 0 - 6, which equals -6.

Key Concepts

Substitution in Algebra

Understanding how to substitute values into algebraic expressions is fundamental to mastering algebra. Substitution means replacing a variable, in this case, x, with a given number to evaluate the expression. For instance, if an expression includes 4x - 6 and we're told to evaluate this expression for x = -1, we directly replace x with -1, leading to a new numerical expression: 4(-1) - 6.

Similarly, when asked to evaluate the same expression for x = 0, substitute 0 for x, leading to 4(0) - 6. It is crucial to substitute precisely and then follow through with the proper arithmetic operations to find the correct value of the expression.

Arithmetic Operations

Once we have substituted the value into the expression, we need to carry out arithmetic operations. These include addition, subtraction, multiplication, and division. In our exercise, after substituting x with -1, we get 4(-1) - 6 which involves multiplication and subtraction. We multiply 4 by -1 to get -4, and then subtract 6 to arrive at -10.

For the second value, x = 0, multiplying 4 by 0 gives 0, and further subtracting 6 results in -6. It's important to perform these operations carefully and in the correct order. In expressions involving multiple operations, always remember to follow the order of operations, commonly known as PEMDAS/BODMAS (Parentheses/Brackets, Orders/power, Division and Multiplication, Addition and Subtraction).

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. In our example, 4x - 6 is an algebraic expression with x as the variable. Evaluating algebraic expressions requires an understanding of both substitution and arithmetic operations. These expressions can represent relationships, patterns, and can be used to solve problems in almost every field of study.

When dealing with algebraic expressions, it's important to familiarize yourself with terms such as coefficients, which are the numerical factors of terms (in our example, 4 is the coefficient of x), constants (like -6), and the variables themselves. Combining these components properly and respecting the arithmetic rules allows us to evaluate the expressions precisely for any given value.