Problem 88 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.. $$Find the missing polynomials: $\quad-\frac{3 x-12}{2 x}=\frac{3}{2}$$

Step-by-Step Solution

Answer

The original statement is true. There are no changes required in order to make it true.

Step 1: Analyze the given equations

The first step is to examine the given equation. It is basically an addition/subtraction of three fractions: \(\frac{2x+1}{x-7}\), \(\frac{3x+1}{x-7}\), and \(\frac{5x+2}{x-7}\). All three fractions have the same denominator, so they can be combined.

Step 2: Combine all fractions

Since the three fractions have the same denominator (\(x-7\)), we can combine these fractions by adding/subtracting the numerators accordingly. This gives: \(\frac{2x+1+3x+1-5x+2}{x-7}\).

Step 3: Simplify the numerator

The next step is to simplify the numerator. Add like terms in the numerator to simplify the fraction. This gives: \(\frac{2x+3x-5x+1+1-2}{x-7} = \frac{0}{x-7}\).

Step 4: Verify the statement

The simplified fraction becomes 0 which is equal to the RHS of the original statement. Hence, the given statement is true. There is no need to modify it to make it true.

Key Concepts

Fraction Addition

When dealing with rational expressions, fraction addition can be challenging if you aren't familiar with the process. Rational expressions are essentially fractions where the numerator and/or the denominator is a polynomial. The rules of fraction addition remain the same as with numerical fractions, meaning we need a common denominator to add them together.
In the original exercise, we have three fractions: \( \frac{2x+1}{x-7} \), \( \frac{3x+1}{x-7} \), and \( \frac{5x+2}{x-7} \). Because they all share the same denominator, \( x-7 \), it simplifies the process. We can directly add the numerators together:

• The sum of the numerators is \( 2x + 1 + 3x + 1 - 5x - 2 \).
• The common denominator remains \( x-7 \).

This merging of numerators is straightforward and only requires attention to the signs and coefficients when combining them.

Simplifying Fractions

Simplifying fractions is an essential skill in algebra, allowing you to reduce expressions and make them easier to handle. In the original exercise, once we combine the fractions, our focus shifts to the numerator of \( \frac{2x+1+3x+1-5x+2}{x-7} \). Simplifying it requires combining like terms:

• Add the coefficients of \(x\) terms: \(2x + 3x - 5x = 0x\).
• Combine the constant terms: \(1 + 1 - 2 = 0\).

This simplification results in a numerator of 0. Thus, the expression becomes \( \frac{0}{x-7} \), which simplifies further to just 0. A fraction with 0 as its numerator is always 0, reminding us of a crucial fact in fraction arithmetic.

Verifying Equations

Verification of an equation is crucial in mathematics to ensure if a statement is true. This involves solving the equation and comparing both sides. In this exercise, we need to confirm whether \( \frac{2x+1}{x-7} + \frac{3x+1}{x-7} - \frac{5x+2}{x-7} = 0 \) holds true.
By simplifying the expression as shown in previous steps, we find \( \frac{0}{x-7} = 0 \). This equality is correct, confirming that the entire equation is true.

• Verification tells us there is no need to adjust the statement, as it is accurate.
• This process ensures reliability in computational solutions in algebra.

Understanding each step in verifying can help avoid errors and reinforces the integrity of algebraic manipulations.