Problem 33
Question
Add or subtract as indicated. $$ \frac{4 x+1}{6 x+5}+\frac{8 x+9}{6 x+5} $$
Step-by-Step Solution
Verified Answer
The sum of the two fractions \(\frac{4x+1}{6x+5} + \frac{8x+9}{6x+5}\) simplifies to \(\frac{6x+5}{3x+2.5}\).
1Step 1: Identify the Like Fractions
In this problem, there are two fractions that already have the same denominator \((6x+5)\). These fractions are \(\frac{4x+1}{6x+5}\) and \(\frac{8x+9}{6x+5}\). So, they are identified as like fractions, which means they can be added directly.
2Step 2: Adding the Numerators
Add the numerators together, \(4x+1\) and \(8x+9\), to get the new numerator. The sum is \((4x+1)+(8x+9) = 12x+10\). The denominator remains the same which is \(6x+5\).
3Step 3: Simplify the Final Answer
The added fractions sum up to \(\frac{12x+10}{6x+5}\). However, this fraction can be simplified. The numerator and denominator share the common factor of \(2\). Dividing both by \(2\), the simplified fraction is \(\frac{6x+5}{3x+2.5}\).
Key Concepts
Like FractionsSimplifying Algebraic ExpressionsAlgebraic Numerators and DenominatorsAlgebra Step-by-Step Problem-Solving
Like Fractions
Understanding like fractions is critical when adding or subtracting algebraic fractions. Like fractions have identical denominators, enabling a straightforward combination of the numerators. For instance, if you encounter \( \frac{a}{c} \) and \( \frac{b}{c} \) with the same denominator \(c\text{,} \) these are considered like fractions.
To add or subtract like fractions, simply perform the operation on the numerators and keep the denominator unchanged. This approach simplifies the process of combining algebraic fractions and forms the foundation for more complex algebraic operations.
To add or subtract like fractions, simply perform the operation on the numerators and keep the denominator unchanged. This approach simplifies the process of combining algebraic fractions and forms the foundation for more complex algebraic operations.
Simplifying Algebraic Expressions
Simplifying algebraic expressions is a fundamental skill in algebra. It involves reducing expressions to their simplest form without changing the original value. This process often includes combining like terms, factoring, and canceling out common factors.
An expression \( \frac{12x + 10}{6x + 5} \) can be simplified by looking for common factors in the numerator and denominator.
An expression \( \frac{12x + 10}{6x + 5} \) can be simplified by looking for common factors in the numerator and denominator.
- First, factor both the numerator and denominator.
- Next, identify any common factors.
- Finally, divide the numerator and the denominator by the common factor, \(2\text{,} \) giving \( \frac{6x + 5}{3x + 2.5} \).
Algebraic Numerators and Denominators
Working with algebraic numerators and denominators requires meticulous attention to detail. Algebraic fractions, akin to numerical fractions, consist of a numerator (top part) and a denominator (bottom part), separated by a fraction bar.
For example, \( \frac{4x+1}{6x+5} \) has \(4x+1\) as the numerator and \(6x+5\) as the denominator. In algebra:
For example, \( \frac{4x+1}{6x+5} \) has \(4x+1\) as the numerator and \(6x+5\) as the denominator. In algebra:
- Numerators and denominators can contain variables, constants, and operators (like addition or subtraction).
- To combine fractions with algebraic numerators and denominators, we first ensure they have a common denominator.
- Once we have like fractions, we add or subtract the numerators accordingly while preserving the common denominator.
Algebra Step-by-Step Problem-Solving
Algebraic problem-solving is markedly easier when approached methodically. A step-by-step approach to solving algebra problems aids in avoiding errors and ensuring comprehension of each stage of the problem. This systematic methodology typically involves several clear steps:
- Identify: Firstly, determine what kind of algebraic problem you are facing and what it is asking you to find or solve.
- Setup: Organize the given information and create an equation or expression that accurately represents the problem.
- Solve: Apply appropriate algebraic techniques such as combining like terms, factoring, expanding, or simplifying.
- Check: After finding a solution, substitute it back into the original equation or expression to verify its accuracy.
Other exercises in this chapter
Problem 33
In Exercises \(31-40,\) factor the difference of two squares. $$36 x^{2}-49$$
View solution Problem 33
In Exercises \(27-38,\) add or subtract terms whenever possible. $$\sqrt{50 x}-\sqrt{8 x}$$
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Simplify each exponential expression $$ \left(x^{-6}\right)^{4} $$
View solution Problem 34
express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression. \(-6\) and 8
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