Problem 70 - Contemporary Precalculus

Question

Simplify, and write the given number without using absolute values. $$|(-2) 3|$$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression $$|(-2) 3|$$ is 6.

Step 1: Evaluate expression within the absolute value function

The given expression is $$|(-2) 3|$$. First, we need to evaluate the multiplication of -2 and 3. $$(-2) \cdot 3 = -6$$.

Step 2: Apply the absolute value function

Now, we have the expression $$|-6|$$. Applying the absolute value function, we find that the absolute value of -6 is equal to 6. So, the final answer is: $$|(-2) 3| = 6$$

Key Concepts

Absolute Value

The concept of absolute value focuses on the distance of a number from zero on the number line. It does not consider direction, meaning it turns any negative number into its positive counterpart. - For example, the absolute value of both -5 and 5 is 5, since they are both 5 units away from zero.When dealing with absolute values in expressions, remember:- If a number inside the absolute value brackets is positive, it remains unchanged.- If it is negative, the absolute value function changes it to its positive form.In the case of the expression \[|(-2) \cdot 3|\],we first perform the multiplication, resulting in -6. Then, the absolute value of -6 is taken, bringing us to the simplified result of 6.

Multiplication of Integers

Multiplication of integers is straightforward, yet knowing the rules is crucial to simplifying expressions correctly. Here are some key points to remember: - When you multiply two integers with different signs, the result is negative. - When both integers have the same sign, whether positive or positive, the result is positive. Applying these rules to our given expression: - We multiply -2 and 3. Since they have different signs, the outcome is -6. Understanding these sign rules helps in quickly and accurately simplifying mathematical expressions, avoiding common mistakes.

Precalculus Problems

Precalculus often involves the simplification and manipulation of expressions. This includes understanding deeply how absolute values and integer operations work, to simplify expressions thoroughly. Here are a few tips when approaching precalculus problems: - Always simplify inside the absolute value first, - Apply integer rules for multiplication and other operations correctly, - Only then handle the absolute values to find the final result. Looking at our original problem: - We first manage the integer multiplication of -2 and 3. After finding this as -6, we apply the absolute value function to simplify. Following these steps ensures a clear, mistake-free approach to precalculus problems.