Problem 112 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The value of \(\frac{|3-7|-2^{3}}{(-2)(-3)}\) is the fraction that results when \(\frac{1}{3}\) is subtracted from \(-\frac{1}{3}\)

Step-by-Step Solution

Answer

The statement is true.

Step 1: Evaluate First Expression

Evaluate the expression \(\frac{|3-7|-2^{3}}{(-2)(-3)}\). First, perform the operation inside the absolute value, that results in \(|-4|\). Absolute value of -4 is 4. Then, perform the exponentiation, resulting in 2^3 = 8. So the numerator becomes \(4 - 8 = -4\). The denominator is a multiplication of negatives, which is positive. So, \((-2)*(-3) = 6\). Therefore, the expression simplifies to \(\frac{-4}{6}\).

Step 2: Simplify Expressions

Simplify the fraction \(\frac{-4}{6}\) by dividing the numerator and the denominator by their greatest common divisor, which is 2. This results in \(\frac{-2}{3}\). Similarly, subtract \(\frac{1}{3}\) from \(-\frac{1}{3}\), which results in -\(\frac{2}{3}\).

Step 3: Compare Results

Compare the results from step 1 and 2. Both expressions evaluated to -\(\frac{2}{3}\). Therefore, the statement is true, and no changes are needed.

Key Concepts

Absolute Value Equations

Understanding absolute value is crucial in algebra as it represents the distance of a number from zero on the number line, which is always non-negative. When solving equations involving absolute values, remember that:

• The absolute value of a positive number is the number itself (e.g., \(|4| = 4\).
• The absolute value of a negative number is the positive version of the number (e.g., \(|-4| = 4\).
• The absolute value of zero is zero (e.g., \(|0| = 0\).

In the original exercise, we handled the absolute value of \(3-7\), which equaled \-4\. Applying the absolute value, \(|-4| = 4\). This concept ensures that all deviations from zero are measured as positive, making it easier to work with expressions involving distance or magnitude.

Exponents

Exponents are a fundamental part of algebra, representing repeated multiplication of a number by itself. For example, \(2^3\) means multiplying 2 by itself three times: \(2 \times 2 \times 2 = 8\). Here are some key points:

• An exponent specifies how many times the base is multiplied by itself.
• Any number raised to the power of zero equals one (e.g., \(5^0 = 1\).
• Multiplying numbers with the same base involves adding their exponents (e.g., \(3^2 \times 3^3 = 3^{2+3} = 3^5\)).

In the exercise, we calculated \(2^3\), resulting in \8\, an essential step for evaluating the expression. By mastering exponents, you can simplify expressions and solve equations more efficiently.

Fractions

Fractions are simply a way to represent parts of a whole, composed of a numerator (top part) and a denominator (bottom part). Key points to remember about fractions include:

• To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).
• Fractions with the same denominator can be added or subtracted directly by their numerators.
• When dividing fractions, multiply by the reciprocal of the divisor.

In the discussed problem, we worked with fractions like \(-\frac{2}{3}\) and simplified expressions by finding the GCD, resulting in \(-\frac{2}{3}\). Understanding fractions is essential for handling various types of algebraic problems and ensures accuracy when performing arithmetic with rational numbers.

True or False Statements

In mathematics, determining whether a statement is true or false is a critical skill. It requires evaluating each part of the statement systematically and comparing the results. Here’s how you can approach them:

• Evaluate all expressions separately before comparing.
• Use reliable calculations and simplifications to ensure accuracy.
• If a statement is false, try to identify how minor adjustments can correct it to become true.

In the exercise, we evaluated two expressions: \(-\frac{2}{3}\) derived from different steps, confirming they were identical. Thus, the original statement was true. Spotting these true or false distinctions efficiently aids in solving mathematical propositions and proving or disproving hypotheses accurately.