11th Grade Q1

Rational inequalities involve expressions in the form of a fraction, where the numerator and denominator are polynomials. These inequalities take the form:

The goal is to determine the values of $x$ that satisfy the inequality.

Step-by-Step Guide

1. Rewrite the Inequality

Ensure the inequality is in standard form. That is in one of the forms

Combine terms if necessary to achieve this form.

2. Find the Critical Points

Critical points occur where:

• The numerator $P(x)=0$ (zeros of the fraction).
• The denominator $Q(x)=0$ (undefined points).

Solve $P(x)=0$ and $Q(x)=0$ to find the critical points. These points divide the number line into intervals.

3. Determine Test Intervals

Using the critical points, divide the real number line into intervals. For example, if the critical points are $x_1,x_2,\dots$, the intervals might be:

4. Test Each Interval

Select a test point from each interval and substitute it into the expression $\frac{P(x)}{Q(x)}$. Determine whether the expression is positive or negative in each interval.

5. Identify the Solution Set

Determine where the inequality holds (positive or negative) based on the test results. Consider the type of inequality:

• Use strict inequalities ($>,<$) to exclude critical points where $P(x)=0$.
• Use non-strict inequalities ($\ge ,\le$) to include critical points where $P(x)=0$, but exclude points where $Q(x)=0$.

Example

Let's consider $\frac{x^2-4}{x-3}>2$.

1. Rewrite the inequality in standard form. Taking $2$ from both sides,

Now simplifying,

2. Find the critical points The zeros of the numerator are critical points. So solve
$x=-2$ is a critical point.
The zeros of the denominator are critical points. So solve
$x=3$ is a critical point.

3. Determine the test intervals: Using our critical points, we split the domain into three test intervals. $(-\infty ,-2)$$(-2,3)$, $(3,\infty )$.

4. Test points: Let's select a test point in each interval.

• $x=-3$: $\frac{(-3)+2}{(-3)-3}=\frac{-2}{-6}=\frac{1}{6}>0$.
• $x=0$: $\frac{0+4}{0-3}=-\frac{2}{3}<0$.
• $x=4$: $\frac{4+2}{4-3}=\frac{6}{1}>0$.

5. Identify the solution set
We are looking for values such that $\frac{x+2}{x-3}>0$. Look at our test points, we get the solution:

Additional Tips

1. Undefined Points: Exclude points where $Q(x)=0$ in all cases.
2. Number Line: Use a number line to visualize test intervals and signs for clarity.
3. Multiple Factors: If $P(x)$ or $Q(x)$ factors further, test each factor’s contribution to the sign in each interval.