Step-by-Step Guide to Solving Polynomial and Rational
Inequalities with Examples

Learn how to solve polynomial and rational inequalities with this step-by-step tutorial. This guide includes clear explanations and detailed example problems to help students master the techniques for analyzing inequality expressions.

Example 1:

Solve the inequality. \[ 2 - \dfrac{7 x}{5} \gt -x + 3 \]

Solution to example 1

Multiply both sides by 5: \[ 10 - 7x > -5x + 15 \] Move all x terms to one side: \[ 10 > 2x + 15 \] Subtract 15 from both sides: \[ -5 > 2x \] Divide by 2: \[ -\frac{5}{2} > x \quad \Rightarrow \quad x < -\frac{5}{2} \]

Final Answer (Interval Form):

\[ (-\infty,\ -\frac{5}{2}) \]

Example 2:

Solve the polynomial inequality. \[ -4 x^2 \gt 4 x - 8 \]

Solution to example 2

Move all terms to one side:

\[ -4x^2 - 4x + 8 > 0 \] Multiply both sides by -1 (reversing the inequality): \[ 4x^2 + 4x - 8 < 0 \] Solve the equation: \[ x^2 + x - 2 = 0 \quad \Rightarrow \quad (x + 2)(x - 1) = 0 \] Analyze the sign of \((x + 2)(x - 1)\):

Find where the expression is zero:

\(x + 2 = 0 \Rightarrow x = -2\)
\(x - 1 = 0 \Rightarrow x = 1\)

These are the critical points that divide the number line into intervals:

\((-\infty, -2)\)
\((-2, 1)\)
\((1, \infty)\)

Test the sign in each interval:

For \(x = -3\) in \((-\infty, -2)\): the expression \( (x + 2)(x - 1) \) is positive.
For \(x = 0\) in \((-2, 1)\): the expression \( (x + 2)(x - 1) \) is negative.
For \(x = 2\) in \((1, \infty)\): the expression \( (x + 2)(x - 1) \) is positive.

Evaluate at critical points:

At \(x = -2\): \((x + 2)(x - 1) = 0\)
At \(x = 1\): \((x + 2)(x - 1) = 0\)

Conclusion: The expression is:

Positive on \((-\infty, -2)\) and \((1, \infty)\)
Zero at \(x = -2\) and \(x = 1\)
Negative on \((-2, 1)\)

We want the expression to be less than 0, so the solution is:

Answer: \[ (-2,\ 1) \]

Example 3:

Solve the rational inequality. \[ \dfrac{1}{x+4} - \dfrac{2}{x-3} \ge 0 \]

Solution to example 3

Get common denominator: \[ \frac{(x - 3) - 2(x + 4)}{(x + 4)(x - 3)} \ge 0 \] Simplify numerator: \[ \frac{-x - 11}{(x + 4)(x - 3)} \ge 0 \] Multiply numerator and denominator by -1 (reversing the inequality): \[ \frac{x + 11}{(x + 4)(x - 3)} \le 0 \] Identify critical points

The expression is zero when numerator is zero: \(x = -11\)

The expression is undefined when denominator is zero: \(x = -4\), \(x = 3\)

Test intervals

Divide number line using critical points: \(-\infty, -11, -4, 3, \infty\)

\[ \begin{array}{|c|c|c|c|c|c|} \hline \textbf{Interval} & \textbf{Test Point} & x+11 & x+4 & x-3 & \text{Sign of Expression} \frac{x + 11}{(x + 4)(x - 3)} \\ \hline (-\infty, -11) & x = -12 & - & - & - & - \\ \hline (-11, -4) & x = -5 & + & - & - & + \\ \hline (-4, 3) & x = 0 & + & + & - & - \\ \hline (3, \infty) & x = 4 & + & + & + & + \\ \hline \end{array} \] Choose intervals where expression is less than or equal to 0

\((-8, -11]\): includes \(x = -11\) where expression is 0
\((-4, 3)\): expression is negative

Final Answer (Interval Notation): \[ (-\infty, -11] \cup (-4, 3) \]

Step-by-Step Guide to Solving Polynomial and Rational Inequalities with Examples