\displaystyle \textbf{Question 1: }\text{Solve each of the following system of inequations in }R.
\displaystyle x+3>0,\hspace{0.3cm}2x<14
\displaystyle \text{Answer:}
\displaystyle \text{Given }x+3>0,\hspace{0.3cm}2x<14
\displaystyle x+3>0\Rightarrow x>-3\qquad\ldots\ldots\ldots\text{i)}
\displaystyle 2x<14\Rightarrow x<7\qquad\ldots\ldots\ldots\text{ii)}
\displaystyle \text{From (i) and (ii), the solution set is }(-3,7).
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Solve each of the following system of inequations in }R.
\displaystyle 2x-7>5-x,\hspace{0.3cm}11-5x\leq1
\displaystyle \text{Answer:}
\displaystyle \text{Given }2x-7>5-x,\hspace{0.3cm}11-5x\leq1
\displaystyle 2x-7>5-x\Rightarrow3x>12\Rightarrow x>4\qquad\ldots\ldots\ldots\text{i)}
\displaystyle 11-5x\leq1\Rightarrow-5x\leq-10\Rightarrow x\geq2\qquad\ldots\ldots\ldots\text{ii)}
\displaystyle \text{From (i) and (ii), the solution set is }(4,\infty).
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{Solve each of the following system of inequations in }R.
\displaystyle x-2>0,\hspace{0.3cm}3x<18
\displaystyle \text{Answer:}
\displaystyle \text{Given }x-2>0,\hspace{0.3cm}3x<18
\displaystyle x-2>0\Rightarrow x>2\qquad\ldots\ldots\ldots\text{i)}
\displaystyle 3x<18\Rightarrow x<6\qquad\ldots\ldots\ldots\text{ii)}
\displaystyle \text{From (i) and (ii), the solution set is }(2,6).
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{Solve each of the following system of inequations in }R.
\displaystyle 2x+6\geq0,\hspace{0.3cm}4x-7<0
\displaystyle \text{Answer:}
\displaystyle \text{Given }2x+6\geq0,\hspace{0.3cm}4x-7<0
\displaystyle 2x+6\geq0\Rightarrow2x\geq-6\Rightarrow x\geq-3\qquad\ldots\ldots\ldots\text{i)}
\displaystyle 4x-7<0\Rightarrow4x<7\Rightarrow x<\frac{7}{4}\qquad\ldots\ldots\ldots\text{ii)}
\displaystyle \text{From (i) and (ii), the solution set is }\left[-3,\frac{7}{4}\right).
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{Solve each of the following system of inequations in }R.
\displaystyle 3x-6>0,\hspace{0.3cm}2x-5>0
\displaystyle \text{Answer:}
\displaystyle \text{Given }3x-6>0,\hspace{0.3cm}2x-5>0
\displaystyle 3x-6>0\Rightarrow3x>6\Rightarrow x>2\qquad\ldots\ldots\ldots\text{i)}
\displaystyle 2x-5>0\Rightarrow2x>5\Rightarrow x>\frac{5}{2}\qquad\ldots\ldots\ldots\text{ii)}
\displaystyle \text{From (i) and (ii), the solution set is }\left(\frac{5}{2},\infty\right).
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{Solve each of the following system of inequations in }R.
\displaystyle 2x-3<7,\hspace{0.3cm}2x>-4
\displaystyle \text{Answer:}
\displaystyle \text{Given }2x-3<7,\hspace{0.3cm}2x>-4
\displaystyle 2x-3<7\Rightarrow2x<10\Rightarrow x<5\qquad\ldots\ldots\ldots\text{i)}
\displaystyle 2x>-4\Rightarrow x>-2\qquad\ldots\ldots\ldots\text{ii)}
\displaystyle \text{From (i) and (ii), the solution set is }(-2,5).
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{Solve each of the following system of inequations in }R.
\displaystyle 2x+5\leq0,\hspace{0.3cm}x-3\leq0
\displaystyle \text{Answer:}
\displaystyle \text{Given }2x+5\leq0,\hspace{0.3cm}x-3\leq0
\displaystyle 2x+5\leq0\Rightarrow2x\leq-5\Rightarrow x\leq-\frac{5}{2}\qquad\ldots\ldots\ldots\text{i)}
\displaystyle x-3\leq0\Rightarrow x\leq3\qquad\ldots\ldots\ldots\text{ii)}
\displaystyle \text{From (i) and (ii), the solution set is }\left(-\infty,-\frac{5}{2}\right].
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{Solve each of the following system of inequations in }R.
\displaystyle 5x-1<24,\hspace{0.3cm}5x+1>-24
\displaystyle \text{Answer:}
\displaystyle \text{Given }5x-1<24,\hspace{0.3cm}5x+1>-24
\displaystyle 5x-1<24\Rightarrow5x<25\Rightarrow x<5\qquad\ldots\ldots\ldots\text{i)}
\displaystyle 5x+1>-24\Rightarrow5x>-25\Rightarrow x>-5\qquad\ldots\ldots\ldots\text{ii)}
\displaystyle \text{From (i) and (ii), the solution set is }(-5,5).
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{Solve each of the following system of inequations in }R.
\displaystyle 3x-1\geq5,\hspace{0.3cm}x+2>-1
\displaystyle \text{Answer:}
\displaystyle \text{Given }3x-1\geq5,\hspace{0.3cm}x+2>-1
\displaystyle 3x-1\geq5\Rightarrow3x\geq6\Rightarrow x\geq2\qquad\ldots\ldots\ldots\text{i)}
\displaystyle x+2>-1\Rightarrow x>-3\qquad\ldots\ldots\ldots\text{ii)}
\displaystyle \text{From (i) and (ii), the solution set is }[2,\infty).
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{Solve each of the following system of inequations in }R.
\displaystyle 11-5x>-4,\hspace{0.3cm}4x+13\leq-11
\displaystyle \text{Answer:}
\displaystyle \text{Given }11-5x>-4,\hspace{0.3cm}4x+13\leq-11
\displaystyle 11-5x>-4\Rightarrow15>5x\Rightarrow x<3\qquad\ldots\ldots\ldots\text{i)}
\displaystyle 4x+13\leq-11\Rightarrow4x\leq-24\Rightarrow x\leq-6\qquad\ldots\ldots\ldots\text{ii)}
\displaystyle \text{From (i) and (ii), the solution set is }(-\infty,-6].
\displaystyle \\

\displaystyle \textbf{Question 11: }\text{Solve each of the following system of inequations in }R.
\displaystyle 4x-1\leq0,\hspace{0.3cm}3-4x<0
\displaystyle \text{Answer:}
\displaystyle \text{Given }4x-1\leq0,\hspace{0.3cm}3-4x<0
\displaystyle 4x-1\leq0\Rightarrow4x\leq1\Rightarrow x\leq\frac{1}{4}\qquad\ldots\ldots\ldots\text{i)}
\displaystyle 3-4x<0\Rightarrow4x>3\Rightarrow x>\frac{3}{4}\qquad\ldots\ldots\ldots\text{ii)}
\displaystyle \text{The solution sets in (i) and (ii) have no common element.}
\displaystyle \therefore \text{The solution set is }\phi.
\displaystyle \\

\displaystyle \textbf{Question 12: }\text{Solve each of the following system of inequations in }R.
\displaystyle x+5>2(x+1),\hspace{0.3cm}2-x<3(x+2)
\displaystyle \text{Answer:}
\displaystyle \text{Given }x+5>2(x+1),\hspace{0.3cm}2-x<3(x+2)
\displaystyle x+5>2(x+1)\Rightarrow x+5>2x+2\Rightarrow x<3\qquad\ldots\ldots\ldots\text{i)}
\displaystyle 2-x<3(x+2)\Rightarrow2-x<3x+6\Rightarrow x>-1\qquad\ldots\ldots\ldots\text{ii)}
\displaystyle \text{From (i) and (ii), the solution set is }(-1,3).
\displaystyle \\

\displaystyle \textbf{Question 13: }\text{Solve each of the following system of inequations in }R.
\displaystyle 2(x-6)<3x-7,\hspace{0.3cm}11-2x<6-x
\displaystyle \text{Answer:}
\displaystyle \text{Given }2(x-6)<3x-7,\hspace{0.3cm}11-2x<6-x
\displaystyle 2(x-6)<3x-7\Rightarrow2x-12<3x-7\Rightarrow x>-5\qquad\ldots\ldots\ldots\text{i)}
\displaystyle 11-2x<6-x\Rightarrow5<x\Rightarrow x>5\qquad\ldots\ldots\ldots\text{ii)}
\displaystyle \text{From (i) and (ii), the solution set is }(5,\infty).
\displaystyle \\

\displaystyle \textbf{Question 14: }\text{Solve each of the following system of inequations in }R.
\displaystyle 5x-7<3(x+3),\hspace{0.3cm}1-\frac{3x}{2}\geq x-4
\displaystyle \text{Answer:}
\displaystyle \text{Given }5x-7<3(x+3),\hspace{0.3cm}1-\frac{3x}{2}\geq x-4
\displaystyle 5x-7<3(x+3)\Rightarrow5x-7<3x+9\Rightarrow x<8\qquad\ldots\ldots\ldots\text{i)}
\displaystyle 1-\frac{3x}{2}\geq x-4\Rightarrow x+\frac{3x}{2}\leq5
\displaystyle \Rightarrow \frac{5x}{2}\leq5\Rightarrow x\leq2\qquad\ldots\ldots\ldots\text{ii)}
\displaystyle \text{From (i) and (ii), the solution set is }(-\infty,2].
\displaystyle \\

\displaystyle \textbf{Question 15: }\text{Solve each of the following system of inequations in }R.
\displaystyle \frac{2x-3}{4}-2\geq\frac{4x}{3}-6,\hspace{0.3cm}2(2x+3)<6(x-2)+10
\displaystyle \text{Answer:}
\displaystyle \text{Given }\frac{2x-3}{4}-2\geq\frac{4x}{3}-6,\hspace{0.3cm}2(2x+3)<6(x-2)+10
\displaystyle \frac{2x-3}{4}-2\geq\frac{4x}{3}-6
\displaystyle \Rightarrow \frac{2x-3-8}{4}\geq\frac{4x-18}{3}
\displaystyle \Rightarrow \frac{2x-11}{4}\geq\frac{4x-18}{3}
\displaystyle \Rightarrow 3(2x-11)\geq4(4x-18)
\displaystyle \Rightarrow 6x-33\geq16x-72
\displaystyle \Rightarrow -10x\geq-39
\displaystyle \Rightarrow x\leq\frac{39}{10}\qquad\ldots\ldots\ldots\text{i)}
\displaystyle 2(2x+3)<6(x-2)+10
\displaystyle \Rightarrow 4x+6<6x-12+10
\displaystyle \Rightarrow 4x+6<6x-2
\displaystyle \Rightarrow -2x<-8
\displaystyle \Rightarrow x>4\qquad\ldots\ldots\ldots\text{ii)}
\displaystyle \text{The solution sets in (i) and (ii) have no common element.}
\displaystyle \therefore \text{The solution set is }\phi.
\displaystyle \\

\displaystyle \textbf{Question 16: }\text{Solve each of the following system of inequations in }R.
\displaystyle \frac{7x-1}{2}<-3,\hspace{0.3cm}\frac{3x+8}{5}+11<0
\displaystyle \text{Answer:}
\displaystyle \text{Given }\frac{7x-1}{2}<-3,\hspace{0.3cm}\frac{3x+8}{5}+11<0
\displaystyle \frac{7x-1}{2}<-3\Rightarrow7x-1<-6\Rightarrow7x<-5
\displaystyle \Rightarrow x<-\frac{5}{7}\qquad\ldots\ldots\ldots\text{i)}
\displaystyle \frac{3x+8}{5}+11<0\Rightarrow3x+8+55<0
\displaystyle \Rightarrow 3x+63<0\Rightarrow3x<-63
\displaystyle \Rightarrow x<-21\qquad\ldots\ldots\ldots\text{ii)}
\displaystyle \text{From (i) and (ii), the solution set is }(-\infty,-21).
\displaystyle \\

\displaystyle \textbf{Question 17: }\text{Solve the following system of inequations in }R:
\displaystyle \frac{2x+1}{7x-1}>5,\hspace{0.3cm}\frac{x+7}{x-8}>2
\displaystyle \text{Answer:}
\displaystyle \text{Given }\frac{2x+1}{7x-1}>5,\hspace{0.3cm}\frac{x+7}{x-8}>2
\displaystyle \frac{2x+1}{7x-1}>5
\displaystyle \Rightarrow \frac{2x+1}{7x-1}-5>0
\displaystyle \Rightarrow \frac{2x+1-5(7x-1)}{7x-1}>0
\displaystyle \Rightarrow \frac{-33x+6}{7x-1}>0
\displaystyle \text{For the fraction to be positive, the numerator and denominator must have the same sign.}
\displaystyle \text{Case I: }-33x+6>0\text{ and }7x-1>0
\displaystyle \Rightarrow x<\frac{2}{11}\text{ and }x>\frac{1}{7}
\displaystyle \Rightarrow \frac{1}{7}<x<\frac{2}{11}
\displaystyle \text{Case II: }-33x+6<0\text{ and }7x-1<0
\displaystyle \Rightarrow x>\frac{2}{11}\text{ and }x<\frac{1}{7}
\displaystyle \text{This is not possible.}
\displaystyle \therefore \text{The solution set of the first inequation is }\left(\frac{1}{7},\frac{2}{11}\right).\qquad\ldots\ldots\ldots\text{i)}
\displaystyle \frac{x+7}{x-8}>2
\displaystyle \Rightarrow \frac{x+7}{x-8}-2>0
\displaystyle \Rightarrow \frac{x+7-2(x-8)}{x-8}>0
\displaystyle \Rightarrow \frac{-x+23}{x-8}>0
\displaystyle \text{For the fraction to be positive, the numerator and denominator must have the same sign.}
\displaystyle \text{Case I: }-x+23>0\text{ and }x-8>0
\displaystyle \Rightarrow x<23\text{ and }x>8
\displaystyle \Rightarrow 8<x<23
\displaystyle \text{Case II: }-x+23<0\text{ and }x-8<0
\displaystyle \Rightarrow x>23\text{ and }x<8
\displaystyle \text{This is not possible.}
\displaystyle \therefore \text{The solution set of the second inequation is }(8,23).\qquad\ldots\ldots\ldots\text{ii)}
\displaystyle \text{The solution sets in (i) and (ii) have no common element.}
\displaystyle \therefore \text{The solution set of the simultaneous inequations is }\phi.
\displaystyle \\

\displaystyle \textbf{Question 18: }\text{Solve each of the following system of inequations in }R.
\displaystyle 0<\frac{-x}{2}<3
\displaystyle \text{Answer:}
\displaystyle \text{Given }0<\frac{-x}{2}<3
\displaystyle \frac{-x}{2}>0\Rightarrow x<0\qquad\ldots\ldots\ldots\text{i)}
\displaystyle \frac{-x}{2}<3\Rightarrow -x<6\Rightarrow x>-6\qquad\ldots\ldots\ldots\text{ii)}
\displaystyle \text{From (i) and (ii), the solution set is }(-6,0).
\displaystyle \\

\displaystyle \textbf{Question 19: }\text{Solve each of the following system of inequations in }R.
\displaystyle 10\leq-5(x-2)<20
\displaystyle \text{Answer:}
\displaystyle \text{Given }10\leq-5(x-2)<20
\displaystyle -5(x-2)\geq10\Rightarrow-5x+10\geq10\Rightarrow-5x\geq0\Rightarrow x\leq0\qquad\ldots\ldots\ldots\text{i)}
\displaystyle -5(x-2)<20\Rightarrow-5x+10<20\Rightarrow-5x<10\Rightarrow x>-2\qquad\ldots\ldots\ldots\text{ii)}
\displaystyle \text{From (i) and (ii), the solution set is }(-2,0].
\displaystyle \\

\displaystyle \textbf{Question 20: }\text{Solve the following system of inequations in }R.
\displaystyle -5<2x-3<5
\displaystyle \text{Answer:}
\displaystyle \text{Given }-5<2x-3<5
\displaystyle -5<2x-3\Rightarrow-2<2x\Rightarrow x>-1\qquad\ldots\ldots\ldots\text{i)}
\displaystyle 2x-3<5\Rightarrow2x<8\Rightarrow x<4\qquad\ldots\ldots\ldots\text{ii)}
\displaystyle \text{From (i) and (ii), the solution set is }(-1,4).
\displaystyle \\

\displaystyle \textbf{Question 21: }\text{Solve }\frac{4}{x+1}\leq3\leq\frac{6}{x+1},\text{ where }x>0.
\displaystyle \text{Answer:}
\displaystyle \frac{4}{x+1}\leq3\leq\frac{6}{x+1},\text{ where }x>0
\displaystyle \text{Since }x>0,\text{ we have }x+1>0.
\displaystyle \therefore 4\leq3(x+1)\leq6
\displaystyle \Rightarrow 4\leq3x+3\leq6
\displaystyle \Rightarrow 1\leq3x\leq3
\displaystyle \Rightarrow \frac{1}{3}\leq x\leq1
\displaystyle \therefore \text{The solution set is }\left[\frac{1}{3},1\right].
\displaystyle \\

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