Question 1: Solve the following inequation and graph the solution on a number line \displaystyle  2x- 5 \leq 5x+4 < 11 , where \displaystyle  x \in I . [2011]

Answer:

\displaystyle  2x-5 \leq 5x+4 < 11  

\displaystyle  2x-5 \leq 5x+4 \text{  or  } -9 \leq 3x \text{  or  } -3 \leq x  

\displaystyle  5x+4 < 11 \text{  or  } 5x < 7 \text{  or  } x < \frac{7}{5}  

\displaystyle  -3 \leq x <\frac{7}{5}  

Therefore \displaystyle  x \in \{-3, -2, -1, 0, 1 \}  

\displaystyle  \\

Question 2: Given that \displaystyle  x \in I , solve the inequation and graph it on a number line: \displaystyle  3 \geq \frac{x-4}{2}+\frac{x}{3} \geq 2 . [2004]

Answer:

\displaystyle  3 \geq \frac{x-4}{2}+\frac{x}{3} \geq 2  

\displaystyle  18 \geq 3(x-4)+2x \geq 12  

\displaystyle  30 \geq 5x \geq 24  

\displaystyle  6 \geq x \geq 4.8  

Therefore \displaystyle  x \in \{5, 6 \}  

\displaystyle  \\

Question 3: Given \displaystyle  A = \{x: 11x-5 > 7x + 3, x \in R \} , \displaystyle  B = \{x: 18x-9 \geq 15+12x , x \in R \} . Find the range of the set \displaystyle  A \cap B and represent it on a number line. [2005]

Answer:

\displaystyle  A: 11x-5 > 7x+3  

\displaystyle  4x >8 \text{  or  } x >2  

\displaystyle  B: 18x-9 \geq 15+12x  

\displaystyle  6x \geq 24 \text{  or  } x \geq 4  

\displaystyle  A \cap B = \{ x: x \geq 4, x \in R \}  

\displaystyle  \\

Question 4: Solve the given inequation and graph it on a number line: \displaystyle  2y-3 < y+1 \leq 4y+7, y \in R . [2008]

Answer:

\displaystyle  2y-3 < y+1 \leq 4y+7  

\displaystyle  2y-3 < y+1 \text{  or  } y < 4  

\displaystyle  y+1 \leq 4y+7 \text{  or  } -6 \leq 3y \text{  or  } -2 \leq y  

Hence \displaystyle  \{ x: -2 \leq y < 4, x \in R \}  

\displaystyle  \\

Question 5: Solve the given inequation and graph it on a number line: \displaystyle  -3 < -\frac{1}{2}-\frac{2x}{3} \leq \frac{5}{6}, x \in R . [2010]

Answer:

\displaystyle  -3 < -\frac{1}{2}-\frac{2x}{3} \leq \frac{5}{6}  

\displaystyle  -3 < -\frac{1}{2}-\frac{2x}{3}  

\displaystyle  -18 < -3 -4x  

\displaystyle  4x < 15 \text{  or  } x < \frac{15}{4}  

\displaystyle  -\frac{1}{2}-\frac{2x}{3} \leq \frac{5}{6} \text{  or  } -3-4x \leq 5  

\displaystyle  -8 \leq 4x \text{  or  } -2 \leq x  

Therefore \displaystyle  \{ x: -2 \leq x < \frac{15}{4}, x \in R \}  

\displaystyle  \\

Question 6: Solve the given inequation and graph it on a number line: \displaystyle  4x-19 < \frac{3x}{5}-2 \leq -\frac{2}{5}+x, x \in R . [2012]

Answer:

\displaystyle  4x-19 < \frac{3x}{5}-2 \leq -\frac{2}{5}+x  

\displaystyle  4x-19 < \frac{3x}{5}-2 \text{  or  } 20x-95 < 3x-10 \text{  or  } 17x < 85 \text{  or  } x < 5  

\displaystyle  \frac{3x}{5}-2 \leq -\frac{2}{5}+x \text{  or  } 3x-10 \leq -2 +5x \text{  or  } -8 \leq 2x \text{  or  } -4 \leq x  

Therefore \displaystyle  \{x : -4 \leq x < 5, x \in R \}  

\displaystyle  \\

\displaystyle \textbf{Question 7. }\text{Solve the given inequality and graph it on a number line:}
\displaystyle -\frac{x}{3}\leq \frac{x}{2}-1-\frac{1}{3}<\frac{x}{6},\ x\in R \hspace{2.2cm} \text{[ICSE 2013]}
\displaystyle \text{Answer:}
\displaystyle -\frac{x}{3}\leq \frac{x}{2}-\frac{4}{3}<\frac{x}{6}
\displaystyle \text{First solve }-\frac{x}{3}\leq \frac{x}{2}-\frac{4}{3}
\displaystyle \Rightarrow -2x\leq 3x-8
\displaystyle \Rightarrow -5x\leq -8
\displaystyle \Rightarrow x\geq \frac{8}{5}
\displaystyle \text{Now solve }\frac{x}{2}-\frac{4}{3}<\frac{x}{6}
\displaystyle \Rightarrow 3x-8<x
\displaystyle \Rightarrow 2x<8
\displaystyle \Rightarrow x<4
\displaystyle \therefore \frac{8}{5}\leq x<4
\displaystyle \text{Hence, the solution set is }\left[\frac{8}{5},4\right)
\displaystyle \text{Number line representation:}


\\

 

Question 8: Find the value of \displaystyle  x which satisfies the inequation: \displaystyle  -2\frac{5}{6} < \frac{1}{2} - \frac{2x}{3} \leq 2, x \in W . [2014]

Answer:

\displaystyle  -2\frac{5}{6} < \frac{1}{2} - \frac{2x}{3} \leq 2  

\displaystyle  -\frac{17}{6} < \frac{1}{2} -\frac{2x}{3} \leq 2  

\displaystyle  -17 < 3-4x \leq 12 \text{  or  } -17 < 3-4x \text{  or  } 4x < 20 \text{  or  } x < 5  

\displaystyle  3-4x \leq 12 \text{  or  } -9 \leq 4x \text{  or  } -2.25 \leq x  

Therefore \displaystyle  \{x : -2.25 \leq x < 5, x \in W \}  

\displaystyle  x \in \{0, 1, 2, 3, 4\}  

Question 9: Solve the inequation: \displaystyle  3-2x \geq x-12 given that \displaystyle  x \in N [1987]

Answer:

\displaystyle  3-2x \geq x-12  

\displaystyle  \Rightarrow 3x \leq 15  

\displaystyle  \Rightarrow x \leq 5 \ or \ x \in \{1, 2, 3, 4, 5 \}  

\displaystyle  \\

Question 10: Solve the inequation: \displaystyle  12+1\frac{5}{6}x \leq 5+3x and \displaystyle  x \in R . [1999]

Answer:

\displaystyle  12+1\frac{5}{6}x \leq 5+3x  

\displaystyle  \Rightarrow 12+\frac{11}{6}x \leq 5+3x  

\displaystyle  \Rightarrow 7 \leq \frac{7}{6}x  

\displaystyle  \Rightarrow x \geq 6 \ or\ \{x: x\in R \ and \ x \geq 6 \}  

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