Since both expressions have a common denominator, x², we can just recopy the denominator and focus on the numerators. We get (9x - 2) - (6x - 8). We must distribute the negative sign over the 6x - 8 expression which gives us 9x - 2 - 6x + 8 ( -2 minus a -8 gives a +6 since a negative and negative make a positive). The numerator is therefore 3x + 6.

Since both fractions in the expression have a common denominator of \(\displaystyle x^{2}\), we can combine like terms into a single numerator over the denominator:

\(\displaystyle \frac{7x-18}{x^{2}}+\frac{6x-14}{x^{2}}\)

\(\displaystyle =\frac{(7x-18)+(6x-14)}{x^{2}}\)

\(\displaystyle =\frac{13x-32}{x^{2}}\)

Since both terms in the expression have the common denominator \(\displaystyle x^{3}\), combine the fractions and simplify the numerators:

\(\displaystyle \frac{10x-9}{x^{3}}+\frac{11x+12}{x^{3}}\)

\(\displaystyle =\frac{(10x-9)+(11x+12)}{x^{3}}\)

\(\displaystyle =\frac{21x+3}{x^{3}}\)

Since both rational terms in the expression have the common denominator \(\displaystyle 2x^{2}\), combine the numerators and simplify like terms:

\(\displaystyle \frac{5x-5}{2x^{2}} + \frac{7x+9}{2x^{2}}\)

\(\displaystyle =\frac{(5x-5)+(7x+9)}{2x^{2}}\)

\(\displaystyle =\frac{12x+4}{2x^{2}}\)

\(\displaystyle =\frac{6x+2}{x^2}\)

When working with complex fractions, it is important not to let them intimidate you. They follow the same rules as regular fractions!

\(\displaystyle \frac{3x^2-4x+5}{x+1}+\frac{4x^2-5x+9}{x+1}\)

In this case, our problem is made easier by the fact that we already have a common denominator. Nothing fancy is required to start. Simply add the numerators:

\(\displaystyle \frac{(3x^2-4x+5)+(4x^2-5x+9)}{x+1}\)

For our next step, we need to combine like terms. This is easier to see if we group them together.

\(\displaystyle \frac{({\color{Blue} 3x^2+4x^2})+({\color{Red} -4x+-5x})+({\color{DarkOrange} 5+9})}{x+1}\)

\(\displaystyle \frac{({\color{Blue}7x^2})+({\color{Red} -9x})+({\color{DarkOrange} 14})}{x+1}\)

Thus, our final answer is:

\(\displaystyle \frac{7x^2 -9x+14}{x+1}\)

Solve for equation for y – x = 3. Then, plug in 3 into (y – x)³ = 27.

Plugging in y=-2 in the second equation, gives x=-13/2. This is the point where the graphs intersect. 

This problem requires the development of an equation.  We are told that the length is 5cm less than 3 times its width.  So we should set up an equation that describes this situation. The equation l = 3w – 5 demonstrates how the length is 5 cm less than 3 times the width of the container.

The highest power this polynomial can achieve is 3.

First, we solve each expression by plugging in the given values for x and y:

3(2²) – 2(3) = 12 – 6 = 6

3(3²) – 2(2) = 27 – 4 = 23

Then we find the difference between the first and second expressions’ values:

23 – 6 = 17