Q1. The solution set for the inequation 2x + 4 ≤ 14, x Ꜫ W is:
(a) {1, 2, 3, 4, 5} (b) {0, 1, 2, 3, 4, 5}
(c) {1, 2, 3, 4} (d) {0, 1, 2, 3, 4} [2023]
Answer: (b) {0,1,2,3,4,5}
Step-by-step Explanation:
\[2x+4\leq14\\2x\leq14-4\\2x\leq10\\x\leq5\\\\Solution\;set\;for\;x\;=\;\{0,1,2,3,4,5\}\\\]
Q2. The solution set of the inequation x – 3 ≥ – 5, x ∊ R is: [1]
(a) {x: x > – 2, x ∊ R} (b) {x: x ≤ -2, x ∊ R}
(c) {x: x ≥ -2, x ∊ R} (d) {-2, -1, 0, 1, 2} [2021 Semester-1]
Answer: (c) {x: x≥ -2, x∊R}
Step-by-step Explanation:
\[x-3\geq-5\\x\geq-5+3\\x\geq-2\\\\Solution\;set\;for\;x\;=\;\{x:\;x\geq-2,\;x\in R\}\\\]
Q3. The solution set on the number line of the linear inequation: [2]
2y – 6 < y + 2 ≤ 2y, y ∈ N is [2021 Semester-1]
Answer: (b)
Step-by-step Explanation:
\[2y-6<y+2\;\;\;;\;y+2\leq2y\\2y-y<2+6\;\;\;;\;y-2y\leq-2\\y<8\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;;\;\;-y\leq-2\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;;\;\;y\geq2\\\\Solution\;set\;for\;x=\{2,3,4,5,6,7\}\]
Q4. Solve the following inequation and represent the solution set on the number line. [2020]
\[\frac{3x}5+2<x+4\leq\frac x2+5.\;\;x\in R\\\]
Step-by-step Explanation:
\[\frac{3x}5+2<x+4\;\;;\;x+4\leq\frac x2+5\\\frac{3x}5-x<4-2\;\;;\;x-\frac x2\leq5-4\\\frac{3x-5x}5<2\;\;;\;\;\frac{2x-x}2\leq1\\-2x<10\;\;\;\;;\;x\leq2\\2x>-10\;\;\;\;;\;x\leq2\\x>-5\;\;\;\;\;\;;\;\;x\leq2\\\\\\Solution\;set\;for\;x=\{x:\;-5<x\leq2,\;x\in R\}\]
Q5. Solve the following in equation and write down the solution set: [3]
11x – 4 < 15x + 4 ≤ 3x + 14, x ∈ W
Represent the solution on a real number line. [2019]
Step-by-step Explanation:
\[11x–4<15x+4\;\;;\;15x+4\leq3x\;+\;14,\\11x-15x<4+4\;\;;\;15x-3x\leq14-4\\-4x<8\;\;;\;12x\leq10\\4x>-8\;\;;\;x\leq\frac{10}{12}\\x>-2\;\;;\;x\leq\frac56\\\\\\Solution\;set\;for\;x=\{\;0\;\}\]
Q6. Solve the following inequation, write down the solution set and represent it on the real number line: [3]
– 2 + 10x ≤ 13x + 10 < 24 + 10x, x ∈ Z [2018]
Step-by-step explanation:
\[–2+10x\leq13x+10\;\;\;;\;13x+10<24+10x\;,\;x\;\in\;Z\\10x-13x\leq10+2\;\;\;;\;13x-10x<24-10\\-3x\leq12\;\;\;;\;3x<14\\3x\geq-12\;\;\;;\;x<\frac{14}3\\x\geq-4\;\;;\;x<4\frac23\\\\\\Solution\;set\;for\;x=\{\;-4,-3,-2,-1,0,1,2,3,4\;\}\]
Q7. Solve the following inequation and represent the solution set on a number line. [3] [2017]
\[-8\frac12<-\frac12-4x\leq7\frac12,\;x\in I\\\]
Step-by-step Explanation:
\[-8\frac12<-\frac12-4x\;\;\;;\;-\frac12-4x\leq7\frac12\\-\frac{17}2+\frac12<-4x\;\;\;;\;-4x\leq\frac{15}2+\frac12\\\frac{-17+1}2<-4x\;\;\;;\;-4x\leq\frac{15+1}2\\\frac{-16}2<-4x\;\;\;;\;-4x\leq\frac{16}2\\-8<-4x\;\;\;;\;-4x\leq8\\8>4x\;\;\;;\;4x\geq-8\\2>x\;\;\;;\;x\geq-2\\\\\\Solution\;set\;for\;x=\{-2,-1,0,1\;\}\]
Q8. Solve the following inequation, write the solution set and represent it on the number line. [2016]
\[-3(x-7)\geq15-7x\;\;\;;\;15-7x>\frac{x+1}3,\;x\in R\\\]
Step-by-step Explanation:
\[-3(x-7)\geq15-7x\;\;\;;\;15-7x>\frac{x+1}3\\-3x+21\geq15-7x\;\;\;;\;3(15-7x)>x+1\\-3x+7x\geq15-21\;\;\;;\;45-21x>x+1\\4x\geq-6\;\;\;;\;-21x-x>1-45\\x\geq\frac{-6}4\;\;\;;\;-22x>-44\\x\geq\frac{-3}2\;\;\;;\;22x<44\\x\geq-1\frac12\;\;\;;\;x<2\\\\\\Solution\;set\;for\;x=\{x:\;-\frac32\leq x<2,\;x\in R\;\}\]
Q9. Solve the following inequation and write the solution set:
13x – 5 < 15x + 4 < 7x + 12, x ε R
Represent the solution on a real number line. [3] [2015]
Step-by-step Explanation:
\[13x–5<15x+4\;\;\;\;;\;\;15x+4<7x+12\\13x-15x<4+5\;\;\;\;;\;\;15x-7x<12-4\\-2x<9\;\;\;\;;\;\;8x<8\\2x>-9\;\;\;\;;\;\;x<1\\x>-\frac92\;\;\;;\;\;x<1\\x>-4\frac12\;\;\;\;;\;\;x<1\\\\\\Solution\;set\;for\;x=\{x:\;-4\frac12<x<1,\;x\in R\;\}\]
Q10. Find the value of x, which satisfy the inequation
\[-2\frac56<\frac12-\frac{2x}3\leq2,\;x\in W.\;\\\]
Graph the solution set on the number line. [3] [2014]
Step-by-step Explanation:
\[-2\frac56<\frac12-\frac{2x}3\;\;\;\;;\;\;\frac12-\frac{2x}3\leq2\;\\\frac{-17}6-\frac12<-\frac{2x}3\;\;\;\;;\;\;-\frac{2x}3\leq2-\frac12\\\frac{-17-3}6<-\frac{2x}3\;\;\;\;;\;\;-\frac{2x}3\leq\frac{4-1}2\\\frac{-20}6<\frac{-2x}3\;\;\;\;;\;\;\frac{-2x}3\leq\frac32\\\frac{-10}3<\frac{-2x}3\;\;\;\;;\;\;-4x\leq9\\-30<-6x\;\;\;\;;\;\;4x\geq-9\\30>6x\;\;\;\;;\;\;x\geq\frac{-9}4\\5>x\;\;\;\;;\;\;x\geq-2\frac14\\\\\\Solution\;set\;for\;x=\{0,1,2,3,4\;\}\]
Q11. Solve the following inequation, write the solution set and represent it on the number line:[2013]
\[-\frac x3\leq\frac x2-1\frac13<\frac16,\;x\in R\\\]
Step-by-step Explanation:
\[-\frac x3\leq\frac x2-1\frac13\;\;\;\;;\;\;\frac x2-1\frac13<\frac16\\\frac{-x}3-\frac x2\leq-\frac43\;\;\;\;;\;\;\frac x2<\frac16+\frac43\\\frac{-2x-3x}6\leq-\frac43\;\;\;\;;\;\;\frac x2<\frac{1+8}6\\\frac{-5x}6\leq\frac{-4}3\;\;\;\;;\;\;6x<18\\-15x\leq-24\;\;\;\;;\;\;x<3\\15x\geq24\;\;\;\;;\;\;x<3\\x\geq\frac{24}{15}\;\;\;\;;\;\;x<3\\x\geq\frac85\;\;\;\;;\;\;x<3\\x\geq1\frac35\;\;\;\;;\;\;x<3\\\\\\Solution\;set\;for\;x=\{\;x:\;\frac85\leq x<3,\;x\in R\;\}\]
Q12. Solve the following inequation and represent the solution set on the number line:[2012]
\[4x-19<\frac{3x}5-2\leq-\frac25+x,\;x\in R\\\]
Step-by-step Explanation:
\[4x-19<\frac{3x}5-2\;\;\;\;;\;\;\frac{3x}5-2\leq-\frac25+x\\4x-\frac{3x}5<-2+19\;\;\;\;;\;\;\frac{3x}5-x\leq-\frac25+2\\\frac{20x-3x}5<17\;\;\;\;;\;\;\frac{3x-5x}5\leq\frac{-2+10}5\\\frac{17x}5<17\;\;\;\;;\;\;\frac{-2x}5\leq\frac85\\17x<85\;\;\;\;;\;\;-10x\leq40\\x<5\;\;\;\;;\;10\;x\geq-40\\x<5\;\;\;\;;\;\;x\geq-4\\\\\\Solution\;set\;for\;x=\{\;x:\;-4\leq x<5,\;x\in R\;\}\]
Q13. Solve the following inequation and represent the solution set on the number line:
2x – 5 ≤ 5x + 4 < 11, where x ∈ I. [3] [2011]
Step-by-step Explanation:
\[2x–5\leq5x+4\;\;\;\;;\;\;5x+4\;<11,\;\\2x-5x\leq4+5\;\;\;\;;\;\;5x<11-4\\-3x\leq9\;\;\;\;;\;\;5x<7\\3x\geq-9\;\;\;\;;\;\;x<\frac75\\x\geq-3\;\;\;\;;\;\;x<1\frac25\\\\\\Solution\;set\;for\;x=\{\;-3,-2,-1,0,1\;\}\]
Q14. Solve the following inequation and represent the solution set on the number line. [2010]
\[-3<-\frac12-\frac{2x}3\leq\frac56\;,x\in R\;\\\]
Step-by-step Explanation:
\[-3<-\frac12-\frac{2x}3\;\;\;\;;\;\;-\frac12-\frac{2x}3\leq\frac56\;\\-3+\frac12<-\frac{2x}3\;\;\;\;;\;\;-\frac{2x}3\leq\frac56\;+\frac12\\\frac{-6+1}2<-\frac{2x}3\;\;\;\;;\;\;-\frac{2x}3\leq\frac{5+3}6\;\\\frac{-5}2<-\frac{2x}3\;\;\;\;;\;\;-\frac{2x}3\leq\frac86\\-15<-4x\;\;\;\;;\;\;-12x\leq24\\15>4x\;\;\;\;;\;\;12x\geq-24\\\frac{15}4>x\;\;\;\;;\;\;x\geq-2\\3\frac34>x\;\;\;\;;\;\;x\geq-2\\\\Solution\;set\;for\;x=\{\;x:\;-2\leq x<3\frac34,\;x\in R\}\]
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