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Chapter 3: Problem 2

Simplify the expressions for the following problems. $$ -|-15| $$

Short Answer

Expert verified
Question: Simplify the expression "-|-15|". Answer: -15

Step by step solution

01

Understand the absolute value function

The absolute value of a number is the distance of that number from zero. The absolute value function is represented by the symbol |x|, where x is the number argument. It is defined as: |a| = a, if a >= 0 and |a| = -a, if a < 0.
02

Find the absolute value of -15

Using the absolute value function definition, we will find the absolute value of -15. |-15| = -(-15) = 15
03

Negate the result

Now that we have found the absolute value of -15, we can proceed to negate the result as indicated by the original expression. -|-15| = -(15) = -15 The simplified expression is -15.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
When you come across a math problem like \(-|-15|\), it's important to break it down into parts that are easier to handle. This is known as simplifying expressions. Simplification helps you make sense of the problem by focusing on one operation at a time.

Start with the innermost part of the expression. Here, you first need to resolve the absolute value of \(-15\). Remember, the absolute value is always non-negative because it represents a distance. Once this part is resolved to \(+15\), the expression becomes clearer.
  • Identify and apply any operations inside symbols like absolute values first.
  • Work through the operations step by step, following the order of operations (parentheses, exponents, multiplication and division, addition and subtraction or PEMDAS).
  • Simplify gradually to make the numbers and operations straightforward.
By simplifying expressions, you make it easier to solve complex mathematical problems correctly.
Negative Numbers
Dealing with negative numbers can be tricky, but understanding them is essential. Negative numbers are values less than zero, often used to indicate direction or loss. In our exercise, once the absolute value of \(-15\) is found to be \(+15\), the negative sign outside the absolute value needs attention.

This external negative sign reverses the positive value you obtained from the absolute step. Simply put, subtracting is like moving left from zero on the number line, whereas adding is like moving right.
  • Negative signs indicate the inverse or opposite direction.
  • If you have a negative sign in front of an expression, it affects the entire value within.
  • Practicing with negatives using number lines can deepen intuition.
Understanding negative numbers helps you manipulate and simplify expressions accurately, even when they seem complicated.
Mathematical Operations
Whenever you tackle expressions like \(-|-15|\), you engage in a series of mathematical operations鈥攅ach with its own rules. These operations include handling absolute values and negatives, as detailed in previous sections. Let's review the structure:

The expression starts with an absolute value \(|-15|\) and results in \(+15\). Then, it moves on to the next operation鈥攁 negation. This changes the problem into \(-15\).
  • Always follow the sequence dictated by the operations, often remembered by PEMDAS.
  • Handle each operation carefully, ensuring the correct sequence to avoid errors.
  • When multiple operations are present, tackle the most contained elements first (like inside absolute value symbols).
Understanding and applying these operations consistently allows you to solve and simplify complex problems effectively in math.

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