Question

You will look for a pattern. What happens to the values of \(\frac{x^{2}+6}{x+2},(x-2),\) and \(\frac{10}{x+2}\) as \(x\) increases?

Short answer

For the expression \(\frac{x^{2}+6}{x+2}\), the value increases with increasing \(x\); for the expression \(x-2\), the value directly increases with increasing \(x\); and for the expression \(\frac{10}{x+2}\), the value decreases with larger \(x\).

Step-by-step solution

Analyzing the first expression \(\frac{x^{2}+6}{x+2}\)

Observe that as \(x\) increases, the value of \(x^{2}\) will increase more significantly than \(x+2\). However, because this is a fraction, the result will be influenced by both the numerator and denominator. The value of \(\frac{x^{2}+6}{x+2}\) will vary with increasing \(x\) but the value of the expression will increase overall.

Analyzing the second expression \(x-2\)

The expression \(x-2\) is linear. As \(x\) increases, the result of the binomial \(x-2\) directly increases as well. Since there are no additional variables or factors involved, this is a straightforward direct relationship.

Analyzing the third expression \(\frac{10}{x+2}\)

In this case, as \(x\) increases, the denominator 'x+2' increases as well. But due to the nature of fractions, as the denominator gets larger, the overall value of the fraction decreases. Thus, the result of \(\frac{10}{x+2}\) will decrease as \(x\) increases.