Problem 112 - Introductory Algebra for College Students

Question

The algebraic expression \(\frac{6 x+6}{x+1}\) cannot have the same value when two different replacements are made for \(x\) such as \(x=-3\) and \(x=2\) 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 algebraic expression \(\frac{6 x+6}{x+1}\) yields different results for \(x=-3\) and \(x=2\). The computed value of \(\frac{|3-7|-2^{3}}{(-2)(-3)}\) is \(-\frac{2}{3}\). The result of the subtraction problem \(-\frac{1}{3} - \(\frac{1}{3}\) is also \(-\frac{2}{3}\).

Step 1: Substitute in the algebraic expression

Substitute \(x=-3\) and \(x=2\) into the algebraic expression \(\frac{6 x+6}{x+1}\). Start with \(x=-3\) to get \(\frac{6(-3)+6}{-3+1} = \frac{-18+6}{-2} = -9\). Then substitute \(x=2\) to get \(\frac{6(2)+6}{2+1} = \frac{12+6}{3} = 6\). So the algebraic expression yields different results for different values of \(x\).

Step 2: Simplify the numerical expression

Evaluate the numerical expression \(\frac{|3-7|-2^{3}}{(-2)(-3)}\). First, calculate the absolute value to get \(|-4|\), which equals \(4\). Next, evaluate the exponent to get \(2^{3} = 8\). Now compute \(\frac{4-8}{6}\) to get \(-\frac{4}{6}\), which simplifies to \(-\frac{2}{3}\).

Step 3: Subtract fractions

Subtract the fraction \(-\frac{1}{3} - \(\frac{1}{3}\). Note that subtracting a positive number is the same as adding a negative number, so the problem becomes \(-\frac{1}{3} + \(-\frac{1}{3}\). Adding these together yields \(-\frac{2}{3}\).

Key Concepts

Substituting Variables in Algebra

Understanding how to substitute variables in algebra is a fundamental skill. When an algebraic expression includes variables like \(x\), you can replace those variables with numbers to compute the value of the expression for specific instances. Think of the variable as a placeholder for the value you're substituting.
For instance, in the given expression \(\frac{6x+6}{x+1}\), you can substitute \(x=-3\) and \(x=2\). To do this accurately, replace the \(x\) with the value you're substituting and carry out the operations. For \(x=-3\), the expression becomes \(\frac{6(-3)+6}{-3+1} = -9\). For \(x=2\), it transforms into \(\frac{6(2)+6}{2+1} = 6\). It's like giving your abstract expression a tangible form to see what different numbers do to its outcome.

• Substitute the value accurately without changing any other part of the expression.
• Perform the operations in the correct order (Order of Operations, or PEMDAS).
• Repeat the process for each new value to compare outcomes.

Simplifying Numerical Expressions

Simplifying numerical expressions is much like putting a complex puzzle together - every operation you perform is a piece connecting to the bigger picture. To simplify an expression such as \(\frac{|3-7|-2^{3}}{(-2)(-3)}\), start by addressing the absolute value and exponent.
First, discern that \(|3-7|\) equals \(4\), since absolute value measures the distance from zero, disregarding negative signs. Then, recognize that \(2^{3}\) is \(8\), which comes from multiplying 2 by itself three times.

Steps for Simplification:

1. Address absolute values and exponents before arithmetic operations.
2. Combine the results: \(4 - 8 = -4\).
3. Divide by \(6\) (resulting from \(-2\) times \(-3\)) to get \(-\frac{2}{3}\).

Tips for Effective Simplification:

• Follow the correct order of arithmetic operations.
• Reduce fractions to their simplest form where possible.
• Keep track of negative signs – they can alter the outcome significantly.

Subtracting Fractions in Algebra

When we subtract fractions in algebra, it's like adjusting pieces on a balance scale to see how they compare. Consider the process of subtracting \(\frac{1}{3}\) from \(-\frac{1}{3}\). This scenario is special because both fractions have the same denominator, which makes the job easier.

Comparable Denominators:

• Keep the denominator the same.
• Subtract the numerators: \((-1) - (+1) = -2\).
• The result is \(-\frac{2}{3}\), which is your new numerator over the common denominator.

Remember that subtracting a positive fraction is equivalent to adding its negative counterpart. Think of negative fractions as owing someone pieces of pie - you're taking away slices, making the amount you owe less.
Here are some helpful points to remember when subtracting fractions with the same denominator:

• Always check if you can simplify the fractions first.
• Keep the denominators constant and subtract the numerators as you would whole numbers.
• Subtracting two negative fractions results in a negative fraction.
• If the fractions are already simplified and have the same denominator, the process is straightforward.