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NCERT Exemplar · Class 11 Mathematics Linear Inequalities

32 questions · 32 still being checked

EXERCISE 6.3 21–32 (part 3 of 3)

  1. Choose the correct answer from the given four options in each of the Exercises $\displaystyle 19$ to $\displaystyle 26$ (M.C.Q.).

    Exercise 21

    If −3x+17<−13\displaystyle -3 x+17<-13, then
    (A)
    x∈(10,∞)\displaystyle x \in(10, \infty)
    (B)
    x∈[10,∞)\displaystyle x \in[10, \infty)
    (C)
    x∈(−∞,10]\displaystyle x \in(-\infty, 10]
    (D)
    x∈[−10,10)\displaystyle x \in[-10,10)

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    NCERT’s answer
    A
    (A) \(\displaystyle x\in(10,\infty)\)\[-3x+17<-13 \]\[-3x<-30 \]Dividing by \(\displaystyle -3\) reverses the sign:\[x>10 \]

  2. Exercise 22

    If x\displaystyle x is a real number and ∣x∣<3\displaystyle |x|<3, then
    (A)
    x≥3\displaystyle x \geq 3
    (B)
    −3<x<3\displaystyle -3<x<3
    (C)
    x≤−3\displaystyle x \leq-3
    (D)
    −3≤x≤3\displaystyle -3 \leq x \leq 3

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    NCERT’s answer
    B
    (B) \(\displaystyle -3<x<3\)\[|x|<3 \ \Leftrightarrow\ -3<x<3 \]The inequality is strict, so \(\displaystyle \pm3\) are excluded.

  3. Exercise 23

    x\displaystyle x and b\displaystyle b are real numbers. If b>0\displaystyle b>0 and ∣x∣>b\displaystyle |x|>b, then
    (A)
    x∈(−b,∞)\displaystyle x \in(-b, \infty)
    (B)
    x∈[−∞,b)\displaystyle x \in[-\infty, b)
    (C)
    x∈(−b,b)\displaystyle x \in(-b, b)
    (D)
    x∈(−∞,−b)∪(b,∞)\displaystyle x \in(-\infty,-b) \cup(b, \infty)

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    NCERT’s answer
    D
    (D) \(\displaystyle x\in(-\infty,-b)\cup(b,\infty)\)For \(\displaystyle b>0\):\[|x|>b \ \Leftrightarrow\ x<-b \ \text{or}\ x>b \]

  4. Exercise 24

    If ∣x−1∣>5\displaystyle |x-1|>5, then
    (A)
    x∈(−4,6)\displaystyle x \in(-4,6)
    (B)
    x∈[−4,6]\displaystyle x \in[-4,6]
    (C)
    x∈[−∞,−4)∪(6,∞)\displaystyle x \in[-\infty,-4) \cup(6, \infty)
    (D)
    x∈[−∞,−4)∪[6,∞)\displaystyle x \in[-\infty,-4) \cup[6, \infty)

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    NCERT’s answer
    C
    (C) \(\displaystyle x\in(-\infty,-4)\cup(6,\infty)\)\[|x-1|>5 \]\[x-1>5 \ \text{or}\ x-1<-5 \]\[x>6 \ \text{or}\ x<-4 \]The inequality is strict, so \(\displaystyle 6\) is not included; this rules out (D).

  5. Exercise 25

    If ∣x+2∣≤9\displaystyle |x+2| \leq 9, then
    (A)
    x∈(−7,11)\displaystyle x \in(-7,11)
    (B)
    x∈[−11,7]\displaystyle x \in[-11,7]
    (C)
    x∈(−∞,−7)∪(11,∞)\displaystyle x \in(-\infty,-7) \cup(11, \infty)
    (D)
    x∈(−∞,−7)∪[11,∞)\displaystyle x \in(-\infty,-7) \cup[11, \infty)

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    NCERT’s answer
    B
    (B) \(\displaystyle x\in[-11,7]\)\[|x+2|\le 9 \iff -9\le x+2\le 9 \]\[-11\le x\le 7 \]

  6. Exercise 26

    The inequality representing the following graph is:
    (A)
    ∣x∣<5\displaystyle |x|<5
    (B)
    ∣x∣≤5\displaystyle |x| \leq 5
    (C)
    ∣x∣>5\displaystyle |x|>5
    (D)
    ∣x∣≥5\displaystyle |x| \geq 5
    NCERT_Question_Class11_Maths_Exemplar_Ch6_Ex6-3_Q26

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    NCERT’s answer
    A
    (A) \(\displaystyle |x|<5\)The shading lies between \(\displaystyle x=-5\) and \(\displaystyle x=5\) with no boundary line drawn, so both edges are excluded.\[-5<x<5 \iff |x|<5 \]
  7. Solution of a linear inequality in variable \(\displaystyle x\) is represented on number line in Exercises $\displaystyle 27$ to 30. Choose the correct answer from the given four options in each of the exercises (M.C.Q.).

    Exercise 27

    (A)
    x∈(−∞,5)\displaystyle x \in(-\infty, 5)
    (B)
    x∈(−∞,5]\displaystyle x \in(-\infty, 5]
    (C)
    x∈[5,∞,)\displaystyle x \in[5, \infty,)
    (D)
    x∈(5,∞)\displaystyle x \in(5, \infty)
    NCERT_Question_Class11_Maths_Exemplar_Ch6_Ex6-3_Q27

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    NCERT’s answer
    D
    (D) \(\displaystyle x\in(5,\infty)\)Open circle at \(\displaystyle 5\) (excluded), ray to the right.\[x>5 \]

  8. Exercise 28

    (A)
    x∈(92,∞)\displaystyle x \in\left(\frac{9}{2}, \infty\right)
    (B)
    x∈[92,∞)\displaystyle x \in\left[\frac{9}{2}, \infty\right)
    (D)
    x∈[−∞,92)\displaystyle x \in\left[-\infty, \frac{9}{2}\right)
    (D)
    x∈(−∞,92]\displaystyle x \in\left(-\infty, \frac{9}{2}\right]
    NCERT_Question_Class11_Maths_Exemplar_Ch6_Ex6-3_Q28

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    NCERT’s answer
    B
    (B) \(\displaystyle x\in\left[\frac{9}{2},\infty\right)\)Filled dot at \(\displaystyle \frac92\) (included), ray to the right.\[x\ge \frac{9}{2} \]

  9. Exercise 29

    (A)
    x∈(−∞,72)\displaystyle x \in\left(-\infty, \frac{7}{2}\right)
    (B)
    x∈(−∞,72]\displaystyle x \in\left(-\infty, \frac{7}{2}\right]
    (C)
    x∈[72,−∞)\displaystyle x \in\left[\frac{7}{2},-\infty\right)
    (D)
    x∈(72,∞)\displaystyle x \in\left(\frac{7}{2}, \infty\right)
    NCERT_Question_Class11_Maths_Exemplar_Ch6_Ex6-3_Q29

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    NCERT’s answer
    A
    (A) \(\displaystyle x\in\left(-\infty,\frac{7}{2}\right)\)Open circle at \(\displaystyle \frac72\) (excluded), ray to the left.\[x<\frac{7}{2} \]

  10. Exercise 30

    (A)
    x∈(−∞,−2)\displaystyle x \in(-\infty,-2)
    (B)
    x∈(−∞,−2]\displaystyle x \in(-\infty,-2]
    (C)
    x∈(−2,∞]\displaystyle x \in(-2, \infty]
    (D)
    x∈[−2,∞)\displaystyle x \in[-2, \infty)
    NCERT_Question_Class11_Maths_Exemplar_Ch6_Ex6-3_Q30

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    NCERT’s answer
    B
    (B) \(\displaystyle x\in(-\infty,-2]\)Filled dot at \(\displaystyle -2\) (included), ray to the left.\[x\le -2 \]

  11. Exercise 31

    State which of the following statements is True or False
    (i)
    If x<y\displaystyle x<y and b<0\displaystyle b<0, then xb<yb\displaystyle \frac{x}{b}<\frac{y}{b}.
    (ii)
    If xy>0\displaystyle x y>0, then x>0\displaystyle x>0 and y<0\displaystyle y<0
    (iii)
    If xy>0\displaystyle x y>0, then x<0\displaystyle x<0 and y<0\displaystyle y<0
    (iv)
    If xy<0\displaystyle x y<0, then x<0\displaystyle x<0 and y<0\displaystyle y<0
    (v)
    If x<−5\displaystyle x<-5 and x<−2\displaystyle x<-2, then x∈(−∞,−5)\displaystyle x \in(-\infty,-5)
    (vi)
    If x<−5\displaystyle x<-5 and x>2\displaystyle x>2, then x∈(−5,2)\displaystyle x \in(-5,2)
    (vii)
    If x>−2\displaystyle x>-2 and x<9\displaystyle x<9, then x∈(−2,9)\displaystyle x \in(-2,9)
    (viii)
    If ∣x∣>5\displaystyle |x|>5, then x∈(−∞,−5)∪[5,∞)\displaystyle x \in(-\infty,-5) \cup[5, \infty)
    (ix)
    If ∣x∣≤4\displaystyle |x| \leq 4, then x∈[−4,4]\displaystyle x \in[-4,4]
    (x)
    Graph of x<3\displaystyle x<3 is
    (xi)
    Graph of x≥0\displaystyle x \geq 0 is
    (xii)
    Graph of y≤0\displaystyle y \leq 0 is
    (xiii) Solution set of x≥0\displaystyle x \geq 0 and y≤0\displaystyle y \leq 0 is
    (xiv) Solution set of x≥0\displaystyle x \geq 0 and y≤1\displaystyle y \leq 1 is
    (xv) Solution set of x+y≥0\displaystyle x+y \geq 0 is
    NCERT_Question_Class11_Maths_Exemplar_Ch6_Ex6-3_Q31
    NCERT_Question_Class11_Maths_Exemplar_Ch6_Ex6-3_Q31_2
    NCERT_Question_Class11_Maths_Exemplar_Ch6_Ex6-3_Q31_3
    NCERT_Question_Class11_Maths_Exemplar_Ch6_Ex6-3_Q31_4
    NCERT_Question_Class11_Maths_Exemplar_Ch6_Ex6-3_Q31_5
    NCERT_Question_Class11_Maths_Exemplar_Ch6_Ex6-3_Q31_6

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    (i) False. Dividing by \(\displaystyle b<0\) reverses the inequality.\[\frac{x}{b}>\frac{y}{b} \](ii) False, (iii) False. Counterexample:\[x=y=1,\quad xy=1>0 \](iv) False.\[x=-1,\ y=1,\quad xy=-1<0 \](v) True.\[(-\infty,-5)\cap(-\infty,-2)=(-\infty,-5) \](vi) False. No \(\displaystyle x\) satisfies both.(vii) True.\[-2<x<9 \iff x\in(-2,9) \](viii) False. At \(\displaystyle x=5\), \(\displaystyle |5|\not>5\).\[|x|>5 \iff x\in(-\infty,-5)\cup(5,\infty) \](ix) True.\[|x|\le 4 \iff -4\le x\le 4 \](x) False. Solid line, so the graph is \(\displaystyle x\le3\).(xi) True. Shading right of the \(\displaystyle y\)-axis, axis included.\[x\ge0 \](xii) False. Shading is above the \(\displaystyle x\)-axis.(xiii) False. Shading is the first quadrant, not the fourth.(xiv) False. Shading is above the line, not below.(xv) True. The line is solid and the shaded side contains \(\displaystyle (1,1)\).\[1+1=2\ge0 \]

  12. Exercise 32

    Fill in the blanks of the following:
    (i)
    If −4x≥12\displaystyle -4 x \geq 12, then x…−3\displaystyle x \ldots-3.
    (ii)
    If −34x≤−3\displaystyle \frac{-3}{4} x \leq-3, then x…4\displaystyle x \ldots 4.
    (iii)
    If 2x+2>0\displaystyle \frac{2}{x+2}>0, then x…−2\displaystyle x \ldots-2.
    (iv)
    If x>−5\displaystyle x>-5, then 4x…−20\displaystyle 4 x \ldots-20.
    (v)
    If x>y\displaystyle x>y and z<0\displaystyle z<0, then −xz…−yz\displaystyle -x z \ldots-y z.
    (vi)
    If p>0\displaystyle p>0 and q<0\displaystyle q<0, then p−q…p\displaystyle p-q \ldots p.
    (vii)
    If ∣x+2∣>5\displaystyle |x+2|>5, then x…−7\displaystyle x \ldots-7 or x…3\displaystyle x \ldots 3.
    (viii)
    If −2x+1≥9\displaystyle -2 x+1 \geq 9, then x…−4\displaystyle x \ldots-4.

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    NCERT’s answer
    (i)
    \(\displaystyle \leq\)
    (ii)
    \(\displaystyle \geq\)
    (iii)
    >
    (iv)
    >
    (v)>(vi)>(vii)<, >
    (viii)
    \(\displaystyle \leq\).
    (i) \(\displaystyle \le\); (ii) \(\displaystyle \ge\); (iii) \(\displaystyle >\); (iv) \(\displaystyle >\); (v) \(\displaystyle >\); (vi) \(\displaystyle >\); (vii) \(\displaystyle <\) and \(\displaystyle >\); (viii) \(\displaystyle \le\)
    (i)
    \[-4x\ge12 \Rightarrow x\le \frac{12}{-4}=-3 \]
    (ii)
    \[\frac{-3}{4}x\le-3 \Rightarrow x\ge -3\cdot\frac{4}{-3}=4 \]
    (iii)
    \[\frac{2}{x+2}>0 \Rightarrow x+2>0 \Rightarrow x>-2 \]
    (iv)
    \[x>-5 \Rightarrow 4x>-20 \]
    (v)
    \[x>y,\ z<0 \Rightarrow xz<yz \Rightarrow -xz>-yz \]
    (vi)
    \[q<0 \Rightarrow -q>0 \Rightarrow p-q>p \]
    (vii)
    \[|x+2|>5 \Rightarrow x+2<-5 \ \text{or}\ x+2>5 \]
    \[x<-7 \ \text{or}\ x>3 \]
    (viii)
    \[-2x+1\ge9 \Rightarrow -2x\ge8 \Rightarrow x\le-4 \]