Problem 49 - Algebra 1

Question

Simplify the expression. $$\frac{22 x+10}{-2}$$

Step-by-Step Solution

Answer

The simplified form of the expression \( \frac{22 x+10}{-2} \) is \( -11x - 5 \).

Step 1: Divide the first term of the numerator

Divide the first term of the numerator (which is \( 22x \)) by the denominator (which is \( -2 \)). This results in \( -11x \).

Step 2: Divide the second term of the numerator

Now, divide the second term of the numerator (which is \( 10 \)) by the denominator (which is \( -2 \)). This results in \( -5 \).

Step 3: Combine the results

Combine the results of step 1 and step 2 to achieve the final simplified form of the original expression, which is \( -11x - 5 \).

Key Concepts

Algebraic Fractions

Algebraic fractions are similar to ordinary fractions, but they contain variables in the numerator, the denominator, or both. To simplify algebraic fractions, you have to perform arithmetic operations while following the variable's rules.

When you encounter an algebraic fraction like \(\frac{22x+10}{-2}\), simplification involves dividing both terms in the numerator by the denominator. It's important to apply the division to each term separately. If you're struggling with this concept, think of the numerator as a mini-expression that needs to be broken down term by term. Once each term is divided by the denominator, you can combine the terms to get the simplified expression.

Here's a tip: always check to see if the numerator and denominator share a common factor. If they do, you can reduce the fraction further by canceling out these common factors, leading to even simpler expressions.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. The expression you're simplifying, \(\frac{22x+10}{-2}\), is one such example. A key part of simplifying rational expressions is to handle one term at a time and keep track of your signs to avoid mistakes.

As you perform the division, be mindful of the negative denominator. It affects each term of the numerator, effectively changing their signs. Negative signs can often be a source of error, so paying close attention to them while simplifying will help ensure your work is accurate.

Remember, the goal is to make the expression easier to work with, which in turn helps with future operations like adding, subtracting, or solving for variables in an equation.

Negative Numbers in Algebra

Dealing with negative numbers is a fundamental part of algebra that can affect the simplification process significantly. When dividing terms by a negative number, the sign of each term changes. This is because dividing by a negative number is the same as multiplying by its positive reciprocal.

In our example, the denominator is \(-2\), so when we divide \(22x\) and \(10\) by \(-2\), each term becomes negative: \(-11x\) and \(-5\), respectively. It's crucial to apply the negative sign to both terms independently to avoid errors. A common pitfall is neglecting to distribute the negative sign correctly, which can lead to the incorrect simplification of the expression.

Always double-check your signs. They can completely change the meaning of an algebraic expression and its potential solutions in an equation. Mastering how to work with negative numbers in algebra will not only help with simplification but also benefit your overall algebraic proficiency.