Solving Polynomial and Rational Inequalities - Pre-Calculus

First, we must solve for the roots of the cubic polynomial equation.

We obtain that the roots are .

Now there are four regions created by these numbers:

• . In this region, the values of the polynomial are negative (i.e.plug in and you obtain

• . In this region, the values of the polynomial are positive (when , polynomial evaluates to )

• . In this region the polynomial switches again to negative.

• . In this region the values of the polynomial are positive

Hence the two regions we want are and .

1. Multiply both sides of the equation by the common denominator of the fractions:

1. Simplify:

1. For standard notation, and the fact that inequalities can be read backwards:

For interval notation:

1. Graph:

Graph the rational expression,

1. Because and a divide by is undefined in the real number system, there is a vertical asymptote where .

2. As , , and as , .

3. As , , and as , .

4. The funtion y is exists over the allowed x-intervals:

One approach for solving the inequality:

For

1. Determine where over the x-values or .

2. for the intervals or .

3. Then the solution is .

Another approach for solving the inequality:

1. Write as , then determine the x-values that cause to be true:

1. is true for or .

2. Then the solution is .

First, subtract from both sides so you get

.

Then find the common denominator and simplify

.

Next, factor out the numerator

and set each of the three factor equal to zero and solve for .

The solutions are

.

Now plug in values between , , , and into the inequality and observe if the conditions of the inequality are met.

Note that . They are met in the interval and .

Thus, the solution to the inequality is

.

We begin by finding the zeros of the equation using the numerator.

So we know that the function will equal zero when . If we just look at the numerator of the function, then this graph would be a parabola with its point at . Now we will solve for the points where the function is undefined by setting the denominator equal to zero and solving for .

And so the function is undefined at . If we make a table to solve for some of the points of the graph:

x	y

And if we graph these points we see something like below (which is our answer). Note that the dotted blue line is the vertical asymptote at .

When solving for a point where the function will be undefined, you set the denominator equal to zero and solve for . This creates a vertical asymptote because when the denominator equals zero the function is undefined and we are solving for . Say for example a function is undefined at . So at all values of where this function is undefined creating a vertical asymptote.

This inequality wants all values where is greater than . So everything up until is included and this is represented by having a dotted line on the graph or an open circle on a number line.

We will first begin by solving for the zeros and undefined points of the inequality. We solve for the zeros by setting the numerator equal to zero.

And so the zeros of this function are at

Now we will solve for the undefined points by setting the denominator equal to zero. Since the denominator is , then whenever , the function is undefined. Now we need to find for which values of is each factor is greater than zero.

For , any value where will be positive and we will be able to graph it. For , any value where will be positive. Now, we can only graph these values up until because the function is undefined here. We are able to pick up the graph again once we reach . The graph will look like the one below.

The zeros of the function are the values of where the function will be equal to zero. In order to find these we set the numerator of the function equal to zero.

We only need to solve for once,

So the zeros of this function are .

To solve for the points at which this function will be undefined, we set the denominator equal to zero and solve for .

And so the function is undefined at

First, we must solve for the roots of the cubic polynomial equation.

We obtain that the roots are .

Now there are four regions created by these numbers:

• . In this region, the values of the polynomial are negative (i.e.plug in and you obtain

• . In this region, the values of the polynomial are positive (when , polynomial evaluates to )

• . In this region the polynomial switches again to negative.

• . In this region the values of the polynomial are positive

Hence the two regions we want are and .

1. Multiply both sides of the equation by the common denominator of the fractions:

1. Simplify:

1. For standard notation, and the fact that inequalities can be read backwards:

For interval notation:

1. Graph:

Graph the rational expression,

1. Because and a divide by is undefined in the real number system, there is a vertical asymptote where .

2. As , , and as , .

3. As , , and as , .

4. The funtion y is exists over the allowed x-intervals:

One approach for solving the inequality:

For

1. Determine where over the x-values or .

2. for the intervals or .

3. Then the solution is .

Another approach for solving the inequality:

1. Write as , then determine the x-values that cause to be true:

1. is true for or .

2. Then the solution is .

First, subtract from both sides so you get

.

Then find the common denominator and simplify

.

Next, factor out the numerator

and set each of the three factor equal to zero and solve for .

The solutions are

.

Now plug in values between , , , and into the inequality and observe if the conditions of the inequality are met.

Note that . They are met in the interval and .

Thus, the solution to the inequality is

.

We begin by finding the zeros of the equation using the numerator.

So we know that the function will equal zero when . If we just look at the numerator of the function, then this graph would be a parabola with its point at . Now we will solve for the points where the function is undefined by setting the denominator equal to zero and solving for .

And so the function is undefined at . If we make a table to solve for some of the points of the graph:

x	y

And if we graph these points we see something like below (which is our answer). Note that the dotted blue line is the vertical asymptote at .

When solving for a point where the function will be undefined, you set the denominator equal to zero and solve for . This creates a vertical asymptote because when the denominator equals zero the function is undefined and we are solving for . Say for example a function is undefined at . So at all values of where this function is undefined creating a vertical asymptote.

This inequality wants all values where is greater than . So everything up until is included and this is represented by having a dotted line on the graph or an open circle on a number line.

We will first begin by solving for the zeros and undefined points of the inequality. We solve for the zeros by setting the numerator equal to zero.

And so the zeros of this function are at

Now we will solve for the undefined points by setting the denominator equal to zero. Since the denominator is , then whenever , the function is undefined. Now we need to find for which values of is each factor is greater than zero.

For , any value where will be positive and we will be able to graph it. For , any value where will be positive. Now, we can only graph these values up until because the function is undefined here. We are able to pick up the graph again once we reach . The graph will look like the one below.

First, we must solve for the roots of the cubic polynomial equation.

We obtain that the roots are .

Now there are four regions created by these numbers:

• . In this region, the values of the polynomial are negative (i.e.plug in and you obtain

• . In this region, the values of the polynomial are positive (when , polynomial evaluates to )

• . In this region the polynomial switches again to negative.

• . In this region the values of the polynomial are positive

Hence the two regions we want are and .

1. Multiply both sides of the equation by the common denominator of the fractions:

1. Simplify:

1. For standard notation, and the fact that inequalities can be read backwards:

For interval notation:

1. Graph: