Evaluating Absolute Value Expressions When T=-12

Jul 8, 2025 by Admin 49 views

Understanding Absolute Value Expressions $-3|t-8|+1.5$ When $t=-12$

In the vast landscape of mathematics, absolute value expressions stand as fundamental building blocks. These expressions, often denoted by vertical bars | |, represent the distance of a number from zero on the number line. Understanding absolute value is crucial for solving a wide array of mathematical problems, from basic algebra to complex calculus. This article delves into the intricacies of absolute value expressions, providing a step-by-step guide to evaluating them and applying them in various contexts.

Decoding the Essence of Absolute Value

The absolute value of a number can be visualized as its magnitude, irrespective of its sign. For instance, the absolute value of 5, denoted as |5|, is simply 5, as it lies 5 units away from zero on the number line. Similarly, the absolute value of -5, denoted as |-5|, is also 5, as it is also 5 units away from zero. This concept can be mathematically expressed as:

|x| = x, if x ≥ 0
|x| = -x, if x < 0

This definition highlights that the absolute value of a non-negative number is the number itself, while the absolute value of a negative number is its positive counterpart.

Evaluating Absolute Value Expressions: A Step-by-Step Approach

Evaluating absolute value expressions involves a systematic process of simplification and computation. Let's consider the expression $-3|t-8|+1.5$ and evaluate it when $t=-12$ . This expression combines absolute value with other arithmetic operations, requiring a careful approach to ensure accuracy.

Step 1: Substitute the Given Value

The first step involves substituting the given value of the variable into the expression. In this case, we replace t with -12:

$-3|-12-8|+1.5$

This substitution sets the stage for the subsequent simplification steps.

Step 2: Simplify the Expression Inside the Absolute Value

Next, we simplify the expression within the absolute value bars. In this case, we subtract 8 from -12:

$-3|-20|+1.5$

This simplification reduces the complexity of the expression within the absolute value.

Step 3: Evaluate the Absolute Value

Now, we evaluate the absolute value of -20, which is the distance of -20 from zero. As per the definition of absolute value, |-20| is 20:

$-3(20)+1.5$

This step transforms the absolute value into its numerical equivalent.

Step 4: Perform Multiplication

Following the order of operations (PEMDAS/BODMAS), we perform the multiplication before addition. We multiply -3 by 20:

$-60+1.5$

This multiplication step further simplifies the expression.

Step 5: Perform Addition

Finally, we perform the addition to obtain the final result. We add 1.5 to -60:

$-58.5$

Therefore, the value of the expression $-3|t-8|+1.5$ when $t=-12$ is -58.5.

The Significance of Order of Operations

The order of operations plays a crucial role in evaluating mathematical expressions, especially those involving absolute values. The acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) serves as a mnemonic to remember the correct sequence. In the context of absolute value expressions, it's essential to simplify the expression within the absolute value bars before applying any operations outside of them. For instance, in the example above, we simplified t - 8 before evaluating the absolute value. Adhering to the order of operations ensures accurate and consistent results.

Exploring Real-World Applications of Absolute Value

Absolute value extends its reach beyond the realm of pure mathematics, finding applications in various real-world scenarios. From calculating distances to measuring errors, absolute value provides a valuable tool for problem-solving. Let's explore some practical examples:

Distance Calculations

In physics and geometry, absolute value is used to determine the distance between two points. For instance, the distance between two points on a number line, say a and b, can be expressed as |a - b|. This ensures that the distance is always a positive value, regardless of the order in which the points are considered.

Error Measurement

In experimental sciences, absolute value helps quantify the difference between an observed value and an expected value. This difference, known as the absolute error, is calculated as |observed value - expected value|. The absolute error provides a measure of the accuracy of an experiment or measurement.

Tolerance in Manufacturing

In manufacturing, absolute value is used to specify the acceptable deviation from a target dimension. For example, a machine part might have a target length of 10 cm with a tolerance of ±0.1 cm. This means the actual length can be anywhere between 9.9 cm and 10.1 cm. The absolute value expression |actual length - target length| ≤ tolerance captures this constraint.

Mastering Absolute Value: Practice Makes Perfect

Understanding and applying absolute value effectively requires practice. Solving a variety of problems involving absolute value expressions will solidify your comprehension and enhance your problem-solving skills. Here are some practice exercises to get you started:

Evaluate the expression $|2x - 5|$ when $x = 3$ .
Simplify the expression $-2|y + 4| + 7$ when $y = -6$ .
Find the distance between the points -7 and 3 on the number line.

By working through these exercises, you'll gain confidence in your ability to manipulate and interpret absolute value expressions.

Conclusion: Absolute Value as a Mathematical Cornerstone

Absolute value, a seemingly simple concept, plays a pivotal role in mathematics and its applications. From evaluating expressions to calculating distances and measuring errors, absolute value provides a versatile tool for problem-solving. By understanding the definition of absolute value, mastering the order of operations, and exploring real-world applications, you can unlock the full potential of this mathematical cornerstone. So, embrace the power of absolute value and embark on a journey of mathematical discovery!

The correct answer is D. -58.5.

What is the value of the expression $-3|t-8|+1.5$ when $t=-12$ ?

Evaluating Absolute Value Expressions When t=-12