Solving Rational Inequality

Rational Inequality

A rational inequality is an inequality that involves a rational expression, which is a ratio of two polynomials. It takes the form: \[\frac{P(x)}{Q(x)} \gt 0 \qquad \frac{P(x)}{Q(x)} \le 0 \] Where,

• \( P(x) and Q(x)\) are polynomials
• We want to find the set of x-values that make the inequality true.

✅ 1. Algebraic Method

Steps

1. Write in standard form: Move everything to one side and simplify.
2. Find critical points (zeros of numerator and denominator).
3. Mark these on a number line.
4. Test intervals between critical points.
5. Pick correct intervals based on the inequality.
6. Pay attention to exclusions (denominator can’t be 0).

Example 1:

Solve: \[\frac{x -3}{x + 2} \gt 0 \] Step 1: Find critical points

• Numerator zero: \(x - 3 = 0 \Rightarrow x = 3\)
• Denominator zero : \( x + 2 = 0 \Rightarrow x = -2\)

Step 2: mark Number line.
Critical points: - 2, 3
Divide into intervals :

• (- ∞ , - 2)
• (-2, 3)
• (3, ∞ )

Step 3: test values

• \(x = - 3 \Rightarrow \frac{-3 -3}{-3 + 2} = \frac{- 6}{- 1} =6 \gt 0 \) ✅
• \(x = 0 \Rightarrow \frac{0 -3}{0 + 2} = \frac{- 3}{+ 2} = - 1.5 \lt 0 \) ❌
• \(x = 4 \Rightarrow \frac{4 -3}{4 - 2} = \frac{1}{2} = 0.5 \gt 0 \) ✅

Step 4: We want where expression is positive: \[✅ (- ∞, - 2 ) ∪ (3, ∞) \] Exclude:

• \(x = 2\) divided by 0 ⛔
• \( x= 3\) makes expression = 0 → not included (because inequality is strict: > 0)

✅ Final Answer: \[(- ∞, - 2 ) ∪ (3, ∞) \]

✅ 2. Graphical Method

Here we graph the function: \[f(x) = \frac{x -3}{x + 2} \gt 0 \] Then find where the graph is above the x-axis (i.e., where \(f(x) \gt 0\) )

1. Plot: \(f(x) = \frac{x -3}{x + 2} \gt 0 \)
2. Find x-intercepts and vertical asymptotes
   • x-intercept\(x = 3\)
   • vertical asymptotes \(x = -2\)
3. Look at intervals:
   • Where is the graph above x-axis? → that's your solution.

✅ Answer from graph:

\[(- ∞, - 2 ) ∪ (3, ∞) \]

Important Notes:

• A rational expression changes sign only at its zeros or undefined points.
• You must exclude points that make the denominator zero.
• A strict inequality (> or<) means you don’t include the zero of numerator.
• A non-strict inequality ( ≥ or ≤ ) means you do include zeros of the numerator, but never include zeros of the denominator.

Example 2: Solve

\[ \frac{x^2 - 9}{x^2 - 4} \le 0 \] Step 1: factor \[ \frac{(x + 3)(x - 3)}{(x + 2)(x - 2)} \le 0 \] Step 2: Critical points
Zeroes:

• Numerator: \(x = -3, 3\)
• Denominator: \(x = -2, 2\)

Number line: mark -3, -2, 2, 3.
Intervals:

• (- ∞, - 3 )
• (- 3 - 2 )
• ( - 2, 2 )
• ( 2, 3 )
• (3, ∞ )

Step 3: Test values

• \(x = - 4: \frac{(+)(−)}{(+)(-)} = (+) \) ❌
• \(x = - 2.5: \frac{(+)(−)}{(+)(+)} = (-) \)✅
• \(x = 0: \frac{(-)(+)}{(+)(+)} = (-) \)✅
• \(x = 2.5: \frac{(-)(+)}{(+)(+)} = (-) \)✅
• \(x = 4: \frac{(+)(−)}{(+)(-)} = (+) \) ❌

Step 4: Select valid intervals where \(\le 0 \)

• (-3, -2), (-2, 2), (2,3)

Also include:

• \(x = -3, 3\) (Numerator zero, inequality is \(\le 0\)) ✅

Exclude:

• \(x = -2, 2\) (denominator zero) ❌

✅ Final Answer:

\[(-3, -2)U(-2, 2)U(2,3) \]

Summary: Principles of Solving Rational Inequalities

🧮 Algebraic Method:

• Factor numerator and denominator.
• Identify critical points (zeros).
• Use number line and test signs in intervals.
• Build solution from intervals where inequality is satisfied.

📊 Graphical Method:

• Plot the rational function.
• dentify:
• x-intercepts (zeros)
• vertical asymptotes (undefined points)
• Find intervals above/below the x-axis.

Interactive Example