Evaluate 8-|2g-5| When G=-3.5 A Step-by-Step Guide

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Introduction

In this article, we will explore how to evaluate the mathematical expression 8 - |2g - 5| when the variable g is equal to -3.5. This involves substituting the given value of g into the expression and simplifying it using the order of operations and the concept of absolute value. Understanding how to evaluate expressions is a fundamental skill in algebra and is crucial for solving more complex mathematical problems. This article aims to provide a clear and detailed explanation of the process, ensuring that readers can confidently tackle similar problems in the future. By breaking down the steps and providing clear explanations, we will demonstrate how to correctly substitute the value, handle the absolute value, and simplify the expression to arrive at the final answer. This exercise not only reinforces basic algebraic skills but also highlights the importance of careful calculation and attention to detail in mathematics. The expression involves both multiplication, subtraction, and absolute value, making it a comprehensive example for understanding order of operations. We will begin by clearly stating the expression and the given value, then proceed with the substitution and simplification steps. Each step will be explained in detail to ensure clarity and understanding. The goal is to not only find the correct answer but also to illustrate the underlying principles and techniques involved in evaluating algebraic expressions.

Understanding the Expression

The expression we need to evaluate is 8 - |2g - 5|. This expression contains several components: a constant (8), a variable (g), multiplication (2 * g), subtraction (both inside and outside the absolute value), and an absolute value function. The absolute value of a number is its distance from zero, so it is always non-negative. For example, |3| = 3 and |-3| = 3. The expression inside the absolute value, 2g - 5, must be evaluated before we can find its absolute value. Then, this absolute value will be subtracted from 8. When g is equal to -3.5, we substitute this value into the expression. This substitution is a critical first step in evaluating any algebraic expression. It involves replacing the variable with its given value, which allows us to move from an expression containing a variable to a numerical expression that can be simplified. After the substitution, we need to follow the order of operations (PEMDAS/BODMAS) to simplify the expression correctly. This order dictates that we perform operations inside parentheses (or absolute value symbols) first, then exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). In our case, this means we first calculate 2 * -3.5, then subtract 5 from the result. After that, we find the absolute value of the outcome, and lastly, subtract this absolute value from 8. Each of these steps is essential for arriving at the correct final answer. Failing to follow the order of operations can lead to incorrect results, highlighting the importance of this fundamental principle in mathematics.

Step-by-Step Evaluation

To evaluate the expression 8 - |2g - 5| when g = -3.5, we will follow these steps:

  1. Substitute the value of g: Replace g with -3.5 in the expression.
    • 8 - |2(-3.5) - 5|
  2. Perform the multiplication inside the absolute value: Multiply 2 by -3.5.
    • 8 - |-7 - 5|
  3. Perform the subtraction inside the absolute value: Subtract 5 from -7.
    • 8 - |-12|
  4. Evaluate the absolute value: Find the absolute value of -12.
    • 8 - 12
  5. Perform the final subtraction: Subtract 12 from 8.
    • -4

Each of these steps is critical to arriving at the correct solution. The initial substitution is straightforward but sets the stage for the rest of the evaluation. Multiplying 2 by -3.5 is a basic arithmetic operation but must be performed accurately to avoid errors later on. Subtracting 5 from -7 requires understanding how to work with negative numbers, which is a fundamental concept in algebra. Taking the absolute value of -12 turns it into 12, as absolute value measures the distance from zero and is always non-negative. Finally, subtracting 12 from 8 involves another operation with negative numbers, resulting in -4. The careful execution of each step, following the correct order of operations, ensures that we arrive at the accurate final answer. It also demonstrates the methodical approach required for solving mathematical problems effectively.

Detailed Calculation

Let's delve into the detailed calculation to ensure clarity and precision. First, we substitute g = -3.5 into the expression:

  • 8 - |2(-3.5) - 5|

Next, we perform the multiplication inside the absolute value:

  • 2 * (-3.5) = -7

So the expression becomes:

  • 8 - |-7 - 5|

Now, we perform the subtraction inside the absolute value:

  • -7 - 5 = -12

Thus, the expression is now:

  • 8 - |-12|

We then evaluate the absolute value of -12, which is the distance of -12 from 0 on the number line. The absolute value of any negative number is its positive counterpart:

  • |-12| = 12

Substituting this back into the expression, we get:

  • 8 - 12

Finally, we perform the subtraction:

  • 8 - 12 = -4

Therefore, the value of the expression 8 - |2g - 5| when g = -3.5 is -4. This detailed step-by-step calculation highlights the importance of careful arithmetic and adherence to the order of operations. Each operation is performed sequentially, ensuring that no mistakes are carried forward. The multiplication and subtraction within the absolute value are handled first, followed by the evaluation of the absolute value itself, and finally, the subtraction from 8. This methodical approach is crucial for solving mathematical problems accurately, particularly those involving multiple operations and negative numbers. The process clearly illustrates how to break down a complex expression into smaller, more manageable steps, making it easier to understand and solve.

Conclusion

In conclusion, the value of the expression 8 - |2g - 5| when g = -3.5 is -4. We arrived at this solution by following a step-by-step approach, which included substituting the given value of g, performing the arithmetic operations inside the absolute value, evaluating the absolute value, and finally, completing the subtraction. This process underscores the importance of adhering to the order of operations (PEMDAS/BODMAS) and being meticulous in each step. Evaluating algebraic expressions is a fundamental skill in mathematics, and this example demonstrates how to apply this skill in a practical context. The methodical approach ensures accuracy and clarity, which are crucial for solving mathematical problems effectively. Understanding how to manipulate expressions, particularly those involving absolute values and negative numbers, is essential for further studies in algebra and other advanced mathematical topics. This exercise also reinforces the concept of absolute value as a measure of distance from zero, which is always non-negative. The negative sign in the final answer comes from subtracting a larger number (12) from a smaller number (8), illustrating the rules of integer arithmetic. By carefully breaking down the problem and executing each step with precision, we successfully evaluated the given expression and found the correct answer. This approach can be applied to a wide range of algebraic problems, making it a valuable tool for any mathematics student.

Final Answer

The final answer is D. -4.