Problem 78 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ 4+\frac{3}{x+2} $$

Step-by-Step Solution

Answer

Based on the above step-by-step solution, the short answer is: The expression $$4 + \frac{3}{x+2}$$ can be rewritten as a single rational expression, which is: $$\frac{4x+11}{x+2}$$

Step 1: Rewrite the whole number as a fraction with common denominator

We need to rewrite the number 4 as a fraction with a denominator (x + 2). We can do this by multiplying 4 by \(\frac{x+2}{x+2}\). This gives us: $$ 4 \cdot \frac{x+2}{x+2} $$ which simplifies to $$ \frac{4(x+2)}{x+2} $$ Now we have two fractions with a common denominator, (x + 2), which we will add together: $$ \frac{4(x+2)}{x+2}+\frac{3}{x+2} $$

Step 2: Add/Subtract the rational expressions

To add the rational expressions, we combine the numerators, keeping the common denominator of (x+2): $$ \frac{4(x+2)+3}{x+2} $$

Step 3: Simplify the numerator and combine like terms if possible

Now we need to simplify the numerator by distributing the "4" and combining like terms: $$ \frac{4x+8+3}{x+2} $$ This simplifies further to: $$ \frac{4x+11}{x+2} $$ The final answer is: $$ \frac{4x+11}{x+2} $$

Key Concepts

Adding Rational Expressions

Adding rational expressions might sound complicated at first, but it's quite similar to adding regular fractions. Imagine you have a pie divided into slices (like fractions) and you want to combine two types of printable slices. Rational expressions are like these pie slices with a little twist—they include variables in the denominators. When adding them, the key is to have the same denominator for each term, just as you would with plain fractions.

For example, let's take a rational expression addition problem, such as adding \( \frac{3}{x+2} \) to a whole number. Since this whole number 4 doesn’t initially share a denominator, we need to rewrite it to fit in with the pie slices family we created in this expression. To do this, we transform 4 into a rational expression by giving it a denominator of \( x+2 \). More on this will be expanded in the next section.

Common Denominator

Finding a common denominator is crucial when dealing with rational expressions. It ensures that each term in the expression shares the same 'bottom' part. Imagine it like trying to stack boxes, where each box has to fit perfectly on top, using its shared base.

In our expression, \( \frac{3}{x+2} \) and the whole number 4, we need to make 4 have a denominator of \( x+2 \) which is currently used by the \( \frac{3}{x+2} \) part, to enable addition. We can achieve this by rewriting 4 as \( \frac{4(x+2)}{x+2} \). This way, both terms in the expression have the \( x+2 \) denominator. Now, they match and can be added together seamlessly! By making sure all parts of an expression share a common denominator, addition or subtraction becomes smooth and manageable.

Simplifying Expressions

Once fractions are combined together over a common denominator, it’s time to simplify the result. This step often involves combining like terms or distributing coefficients in the numerator. Think of it like tidying up after crafting, where you glue pieces of paper (terms) neatly together to create one beautiful piece.

Following our example, we end up with \( \frac{4(x+2) + 3}{x+2} \). The next step is to simplify the numerator. By distributing 4, we expand it to \( 4x + 8 \). Adding the 3 from the other fraction results in the simplified numerator \( 4x + 11 \). So, the expression neatly tidies up to \( \frac{4x + 11}{x+2} \).

Thus, simplifying expressions means organizing terms to their most compact and understandable form. It's the final piece where all pieces of your pie become one perfect round circle ready to be enjoyed!