earn to solve rational inequalities by studying the sign of numerator and denominator, building the sign chart, and writing the solution intervals correctly.
Exercise 1 — level ★★☆☆☆
\[ \frac{x-1}{x+3} > 0 \]
Answer
\[ x < -3 \quad \text{or} \quad x > 1 \]
Solution
Domain conditions
\(x \neq -3\).
Sign chart
Solution set
\[ S = (-\infty,\,-3)\cup(1,\,+\infty) \]
Answer
\[ \boxed{x < -3 \quad \text{or} \quad x > 1} \]
Exercise 2 — level ★★☆☆☆
\[ \frac{3x}{x-4} \leq 0 \]
Answer
\[ 0 \leq x < 4 \]
Solution
Domain conditions
\(x \neq 4\).
Sign chart
Solution set
\[ S = [0,\,4) \]
Answer
\[ \boxed{0 \leq x < 4} \]
Exercise 3 — level ★★☆☆☆
\[ \frac{x+4}{2x-6} \geq 0 \]
Answer
\[ x \leq -4 \quad \text{or} \quad x > 3 \]
Solution
Domain conditions
\(x \neq 3\).
Sign chart
Solution set
\[ S = (-\infty,\,-4]\cup(3,\,+\infty) \]
Answer
\[ \boxed{x \leq -4 \quad \text{or} \quad x > 3} \]
Exercise 4 — level ★★☆☆☆
\[ \frac{5-x}{x+1} < 0 \]
Answer
\[ x < -1 \quad \text{or} \quad x > 5 \]
Solution
Domain conditions
\(x \neq -1\).
Sign chart
Solution set
\[ S = (-\infty,\,-1)\cup(5,\,+\infty) \]
Answer
\[ \boxed{x < -1 \quad \text{or} \quad x > 5} \]
Exercise 5 — level ★★★☆☆
\[ \frac{x^2-4x}{x+2} \geq 0 \]
Answer
\[ -2 < x \leq 0 \quad \text{or} \quad x \geq 4 \]
Solution
Factorisation
\[ \frac{x(x-4)}{x+2} \geq 0 \]
Restriction: \(x\neq-2\).
Sign chart
Solution set
\[ S = (-2,\,0]\cup[4,\,+\infty) \]
Answer
\[ \boxed{-2 < x \leq 0 \quad \text{or} \quad x \geq 4} \]
Exercise 6 — level ★★★☆☆
\[ \frac{x^2-9}{x-1} \leq 0 \]
Answer
\[ x \leq -3 \quad \text{or} \quad 1 < x \leq 3 \]
Solution
Factorisation
\[ \frac{(x-3)(x+3)}{x-1} \leq 0 \]
Restriction: \(x\neq1\).
Sign chart
Solution set
\[ S = (-\infty,\,-3]\cup(1,\,3] \]
Answer
\[ \boxed{x \leq -3 \quad \text{or} \quad 1 < x \leq 3} \]
Exercise 7 — level ★★★☆☆
\[ \frac{2x-1}{x+2} < 3 \]
Answer
\[ x < -7 \quad \text{or} \quad x > -2 \]
Solution
Rewriting
\[ \frac{2x-1}{x+2}-3 < 0 \implies \frac{-x-7}{x+2} < 0 \implies \frac{x+7}{x+2} > 0 \]
Sign chart
Solution set
\[ S = (-\infty,\,-7)\cup(-2,\,+\infty) \]
Answer
\[ \boxed{x < -7 \quad \text{or} \quad x > -2} \]
Exercise 8 — level ★★★☆☆
\[ \frac{x^2+x}{x-2} > 0 \]
Answer
\[ -1 < x < 0 \quad \text{or} \quad x > 2 \]
Solution
Factorisation
\[ \frac{x(x+1)}{x-2} > 0 \]
Restriction: \(x\neq2\).
Sign chart
Solution set
\[ S = (-1,\,0)\cup(2,\,+\infty) \]
Answer
\[ \boxed{-1 < x < 0 \quad \text{or} \quad x > 2} \]
Exercise 9 — level ★★★☆☆
\[ \frac{x+3}{x^2-1} \leq 0 \]
Answer
\[ x \leq -3 \quad \text{or} \quad -1 < x < 1 \]
Solution
Factorisation
\[ \frac{x+3}{(x-1)(x+1)} \leq 0 \]
Restriction: \(x\neq\pm1\).
Sign chart
Solution set
\[ S = (-\infty,\,-3]\cup(-1,\,1) \]
Answer
\[ \boxed{x \leq -3 \quad \text{or} \quad -1 < x < 1} \]
Exercise 10 — level ★★★☆☆
\[ \frac{x^2-2x-8}{x+5} > 0 \]
Answer
\[ -5 < x < -2 \quad \text{or} \quad x > 4 \]
Solution
Factorisation
\[ \frac{(x-4)(x+2)}{x+5} > 0 \]
Restriction: \(x\neq-5\).
Sign chart
Solution set
\[ S = (-5,\,-2)\cup(4,\,+\infty) \]
Answer
\[ \boxed{-5 < x < -2 \quad \text{or} \quad x > 4} \]
Exercise 11 — level ★★★★☆
\[ \frac{x-2}{x+1} \geq \frac{x}{x-3} \]
Answer
\[ x < -1 \quad \text{or} \quad 1 \leq x < 3 \]
Solution
Rewriting
\[ \frac{(x-2)(x-3)-x(x+1)}{(x+1)(x-3)} \geq 0 \implies \frac{-6(x-1)}{(x+1)(x-3)} \geq 0 \implies \frac{x-1}{(x+1)(x-3)} \leq 0 \]
Sign chart
Solution set
\[ S = (-\infty,\,-1)\cup[1,\,3) \]
Answer
\[ \boxed{x < -1 \quad \text{or} \quad 1 \leq x < 3} \]
Exercise 12 — level ★★★★☆
\[ \frac{2}{x+2} - \frac{1}{x-1} \geq 0 \]
Answer
\[ -2 < x < 1 \quad \text{or} \quad x \geq 4 \]
Solution
Rewriting
\[ \frac{2(x-1)-(x+2)}{(x+2)(x-1)} \geq 0 \implies \frac{x-4}{(x+2)(x-1)} \geq 0 \]
Sign chart
Solution set
\[ S = (-2,\,1)\cup[4,\,+\infty) \]
Answer
\[ \boxed{-2 < x < 1 \quad \text{or} \quad x \geq 4} \]
Exercise 13 — level ★★★★☆
\[ \frac{x^2-x-2}{x^2+3x} < 0 \]
Answer
\[ -3 < x < -1 \quad \text{or} \quad 0 < x < 2 \]
Solution
Factorisation
\[ \frac{(x-2)(x+1)}{x(x+3)} < 0 \]
Restriction: \(x\neq0\), \(x\neq-3\).
Sign chart
Solution set
\[ S = (-3,\,-1)\cup(0,\,2) \]
Answer
\[ \boxed{-3 < x < -1 \quad \text{or} \quad 0 < x < 2} \]
Exercise 14 — level ★★★★☆
\[ \frac{3x+6}{x^2-4x+4} > 0 \]
Answer
\[ x > -2 \quad \text{with} \quad x \neq 2 \]
Solution
Factorisation
\[ \frac{3(x+2)}{(x-2)^2} > 0 \]
Remark
The denominator \((x-2)^2\) is always positive for \(x\neq2\): the sign depends solely on the numerator.
Sign chart
Solution set
\[ S = (-2,\,2)\cup(2,\,+\infty) \]
Answer
\[ \boxed{x > -2 \quad \text{with} \quad x \neq 2} \]
Exercise 15 — level ★★★★☆
\[ \frac{x^2-4}{x^2-x-6} \leq 0 \]
Answer
\[ 2 \leq x < 3 \]
Solution
Simplification
\[ \frac{(x-2)(x+2)}{(x-3)(x+2)} = \frac{x-2}{x-3} \quad (x\neq-2) \]
Restriction: \(x\neq3\), \(x\neq-2\).
Sign chart of the reduced form
The reduced form is \(\leq 0\) on \([2,\,3)\). The region \(x<2\) yields a positive fraction; \(x=-2\) is excluded by the domain conditions.
Answer
\[ \boxed{2 \leq x < 3} \]
Exercise 16 — level ★★★★★
\[ \frac{x^2-3x+2}{x^2-2x-3} \geq 0 \]
Answer
\[ x < -1 \quad \text{or} \quad 1 \leq x \leq 2 \quad \text{or} \quad x > 3 \]
Solution
Factorisation
\[ \frac{(x-1)(x-2)}{(x-3)(x+1)} \geq 0 \]
Restriction: \(x\neq3\), \(x\neq-1\). Critical points: \(-1,\,1,\,2,\,3\).
Sign chart
Solution set
\[ S = (-\infty,\,-1)\cup[1,\,2]\cup(3,\,+\infty) \]
Answer
\[ \boxed{x < -1 \quad \text{or} \quad 1 \leq x \leq 2 \quad \text{or} \quad x > 3} \]
Exercise 17 — level ★★★★★
\[ \frac{1}{x} - \frac{1}{x-2} + \frac{1}{x-4} < 0 \]
Answer
\[ x < 0 \quad \text{or} \quad 2 < x < 4 \]
Solution
Rewriting
\[ \frac{(x-2)(x-4)-x(x-4)+x(x-2)}{x(x-2)(x-4)} = \frac{x^2-4x+8}{x(x-2)(x-4)} < 0 \]
Discriminant of the numerator
\[ \Delta = 16-32 = -16 < 0 \implies x^2-4x+8 > 0 \quad \forall\, x \]
The sign of the fraction depends solely on the denominator.
Sign chart
Solution set
\[ S = (-\infty,\,0)\cup(2,\,4) \]
Answer
\[ \boxed{x < 0 \quad \text{or} \quad 2 < x < 4} \]
Exercise 18 — level ★★★★★
\[ \left|\frac{x+1}{x-2}\right| < 3 \]
Answer
\[ x < \dfrac{5}{4} \quad \text{or} \quad x > \dfrac{7}{2} \]
Solution
Eliminating the absolute value
\[ -3 < \frac{x+1}{x-2} < 3 \]
Right condition: \(\dfrac{x+1}{x-2} < 3 \implies \dfrac{2x-7}{x-2} > 0\)
Solution: \(x < 2\) or \(x > 7/2\).
Left condition: \(\dfrac{x+1}{x-2} > -3 \implies \dfrac{4x-5}{x-2} > 0\)
Solution: \(x < 5/4\) or \(x > 2\).
Intersection via sign chart of \(\dfrac{(4x-5)(2x-7)}{(x-2)^2} > 0\)
Solution set
\[ S = \left(-\infty,\,\tfrac{5}{4}\right)\cup\left(\tfrac{7}{2},\,+\infty\right) \]
Answer
\[ \boxed{x < \frac{5}{4} \quad \text{or} \quad x > \frac{7}{2}} \]
Exercise 19 — level ★★★★★
\[ \frac{x^3-x}{x^2-4} \geq 0 \]
Answer
\[ -2 < x \leq -1 \quad \text{or} \quad 0 \leq x \leq 1 \quad \text{or} \quad x > 2 \]
Solution
Factorisation
\[ \frac{x(x-1)(x+1)}{(x-2)(x+2)} \geq 0 \]
Restriction: \(x\neq\pm2\). Critical points: \(-2,\,-1,\,0,\,1,\,2\).
Sign chart
Solution set
\[ S = (-2,\,-1]\cup[0,\,1]\cup(2,\,+\infty) \]
Answer
\[ \boxed{-2 < x \leq -1 \quad \text{or} \quad 0 \leq x \leq 1 \quad \text{or} \quad x > 2} \]
Exercise 20 — level ★★★★★
\[ \frac{x^2-x-6}{x^2-4x+3} \leq \frac{1}{2} \]
Answer
\[ -5 \leq x < 1 \]
Solution
Rewriting
\[ \frac{2(x^2-x-6)-(x^2-4x+3)}{2(x^2-4x+3)} \leq 0 \implies \frac{x^2+2x-15}{2(x-1)(x-3)} \leq 0 \]
Factorisation of the numerator
\[ x^2+2x-15=(x+5)(x-3) \]
The factor \((x-3)\) cancels out (for \(x\neq3\)):
\[ \frac{(x+5)(x-3)}{2(x-1)(x-3)} = \frac{x+5}{2(x-1)} \implies \frac{x+5}{x-1} \leq 0 \]
Restriction: \(x\neq1\), \(x\neq3\). Zeros: \(x=-5\), \(x=1\).
Sign chart
Solution set
\[ S = [-5,\,1) \]
Answer
\[ \boxed{-5 \leq x < 1} \]