Answers to Polynomial and Rational Inequalities

This page presents the correct answers to the questions found in Polynomial and Rational Inequalities – Questions and Answers . All inequalities are written using proper mathematical notation, and solution sets are expressed using interval notation.

Question 1

Solve the inequality:

\[ -(x + 2) + 2x > 2(x - 3) + 3x \]

Answer:

\[ (-\infty,\,1) \]

Question 2

Solve the polynomial inequality:

\[ (6x + 2)(10 - 20x) \ge 0 \]

Answer:

\[ \left[-\frac{1}{3},\,\frac{1}{2}\right] \]

Question 3

Solve the inequality:

\[ x^2 + 3x - 4 > 0 \]

Answer:

\[ (-\infty,\,-4)\;\cup\;(1,\,+\infty) \]

Question 4

Solve the inequality:

\[ 2x^2 < 32 \]

Answer:

\[ (-4,\,4) \]

Question 5

Solve the inequality:

\[ 25x^2 \le 9x \]

Answer:

\[ \left[0,\,\frac{9}{25}\right] \]

Question 6

Solve the inequality:

\[ \frac{1 + x}{|x - 2|} > 0 \]

Answer:

\[ (-1,\,2)\;\cup\;(2,\,+\infty) \]

Question 7

Solve the inequality:

\[ \frac{(2x^2 + 3)(x + 2)}{x^2 - 4} \ge 0 \]

Answer:

\[ (2,\,+\infty) \]

Question 8

Solve the inequality:

\[ \frac{4}{6x - 4} \le \frac{2}{2x + 2} \]

Answer:

\[ (-1,\,\tfrac{2}{3})\;\cup\;[4,\,+\infty) \]

Question 9

Solve the compound inequality:

\[ 2 < \frac{|x + 1|}{2} - 2 < 5 \]

Answer:

\[ (-15,\,-9)\;\cup\;(7,\,13) \]

Question 10

Solve the inequality:

\[ x^2 < -x^4 \]

Answer:

\[ \text{No solution} \]

Related topic: Solving Inequalities in One Variable