Problem 19 - Precalculus

Question

Evaluate each expression, given that \(a=-2\) \(b=3,\) and \(c=-4\). $$|c|-|b|-|a|$$

Step-by-Step Solution

Answer

-1

Step 1: Find the Absolute Value of Each Variable

Calculate the absolute values for each variable: 1. Find \(|c|\): Since \(c = -4\), \(|c| = 4\).2. Find \(|b|\): Since \(b = 3\), \(|b| = 3\).3. Find \(|a|\): Since \(a = -2\), \(|a| = 2\).

Step 2: Substitute the Absolute Values into the Expression

Substitute the absolute values we found in Step 1 into the expression \(|c| - |b| - |a|\):\[|c| - |b| - |a| = 4 - 3 - 2\]

Step 3: Evaluate the Expression

Perform the subtraction operations:1. Subtract \(3\) from \(4\): \[4 - 3 = 1\]2. Subtract \(2\) from the result: \[1 - 2 = -1\]

Step 4: Conclusion: Final Answer

The evaluated expression \(|c| - |b| - |a|\) simplifies to \(-1\).

Key Concepts

Evaluating Expressions

When we talk about evaluating expressions, it essentially means calculating the value of a mathematical expression. In this context, using given numerical values, we follow specific instructions to arrive at the final result. Evaluating an expression often involves:

• Substituting known values into the expression.
• Performing arithmetic operations like addition, subtraction, multiplication, and division.

For instance, in the original exercise, we had to evaluate \[|c|-|b|-|a|\]given that \( a = -2 \), \( b = 3 \), and \( c = -4 \). Evaluating this expression involved calculating the absolute values of each variable and then substituting these values back into the expression before performing the arithmetic operations of subtraction.

Understanding how to evaluate such expressions lays the foundation for solving more complex mathematical problems.

Substitution

Substitution is a method used to solve expressions where specific values replace variables. This technique simplifies the expression, making it easier to compute. To substitute means to:

• Take the known values associated with variables.
• Plug these values into the equation or expression.

In our problem, substitution meant taking the absolute values we calculated:

• \(|a|=2\)
• \(|b|=3\)
• \(|c|=4\)

And replacing \(|a|\), \(|b|\), and \(|c|\) in the expression \(|c| - |b| - |a|\). This step is crucial as it translates abstract expressions into solvable arithmetic operations. Through substitution, expressions become concrete, allowing straightforward evaluation and solution.

Precalculus Mathematics

Precalculus is a branch of mathematics that prepares students for calculus. It includes concepts like functions, algebra, trigonometry, and complex numbers. One vital component of precalculus is understanding and manipulating absolute values and expressions.
Precalculus teaches us how to handle absolute values, which represent the distance of a number from zero on a number line. This concept is fundamental, as shown in our problem where the absolute values of \( a = -2 \), \( b = 3 \), and \( c = -4 \) were pivotal in forming a solvable expression.
Moreover, precalculus emphasizes:

• Working with algebraic expressions and equations.
• Utilizing substitution to transform expressions.
• Developing skills for problem-solving, such as addressing the expression \(|c| - |b| - |a|\).

Grasping these concepts in precalculus is crucial as they are extensively used in higher-level mathematics and real-world applications.