SOLUTION: Considering \[x\text{ }+\text{ }y\text{ }=\text{ }8,\] The shaded region and the origin both are on the same side of the graph of the line and (0, 0) satisfy the constraint \[x\text{...
Solution: For above fraction be greater than 0, either both denominator and numerator should be greater than 0 or both should be less than 0. \[\begin{array}{*{35}{l}} \Rightarrow ~6\text{ }\text{...
\[T\text{ }=\text{ }30\text{ }+\text{ }25\left( x\text{ }\text{ }3 \right),\text{ }3\text{ }\le \text{ }x\text{ }\le \text{ }15;\] where, T = temperature and x = depth inside the earth The...
Let the length of shortest side = ‘x’ cm According to the question, The longest side of a triangle is twice the shortest side ⇒ Length of largest side = 2x Also, the third side is 2 cm longer than...
Let temperature in Celsius be C Let temperature in Fahrenheit be F According to the question, Solution should be kept between 40° C and 45°C ⇒ 40 < C < 45 Multiplying each term by 9/5, we get...
According to the question, Let x litres of 3% solution is to be added to 460 liters of the 9% of solution Then, we get, Total solution = \[\left( 460\text{ }+\text{ }x \right)\text{ }litres\] Total...
According to the question, First reading = 8.48 Second reading = 8.35 Now, let the third reading be ‘x’ Average pH should be between 8.2 and 8.5 Average \[pH~=\text{ }\left( 8.48\text{ }+\text{...
Since, Profit = Revenue – cost Requirement is, profit > 0 \[C\left( x \right)\text{ }=\text{ }26,000\text{ }+\text{ }30\text{ }x;\] where x is number of cassettes \[\begin{array}{*{35}{l}}...
\[\begin{array}{*{35}{l}} 4x\text{ }+\text{ }3\text{ }\ge \text{ }2x\text{ }+\text{ }17 \\ \Rightarrow ~4x\text{ }\text{ }-2x\text{ }\ge \text{ }17\text{ }\text{ }-3 \\ \Rightarrow ~2x\text{ }\ge...
Solution: Multiplying each term by 4, we get \[\Rightarrow ~-20\text{ }\le \text{ }2\text{ }\text{ }-3x\text{ }\le \text{ }36\] Adding -2 each term, we get \[\Rightarrow ~-22\text{ }\le \text{...
\[\left| x\text{ }\text{ }-1 \right|\le \text{ }5\] There are two cases, 1:-\[x\text{ }\text{ }-1\text{ }\le \text{ }5\] Adding 1 to LHS and RHS \[\begin{array}{*{35}{l}} \Rightarrow ~x\text{ }\le...
Solution: \[\begin{array}{*{35}{l}} \Rightarrow ~5-\text{ }\text{ }\left| x \right|\text{ }\le \text{ }0\text{ }and\text{ }\left| x \right|\text{ }\text{ }-3\text{ }>\text{ }0\text{ }or\text{...
SOLUTION: Multiplying each term by \[\begin{array}{*{35}{l}} \left( x\text{ }+\text{ }1 \right) \\ \Rightarrow ~4\text{ }\le \text{ }3\left( x\text{ }+\text{ }1 \right)\text{ }\le \text{ }6 \\...
Solution: The given inequalities are \[x+2y\le 10\], \[x+y\ge 1\], \[x-y\le 0\], \[x\ge 0,y\ge 0\] For \[x+2y\le 10\], Let us put the value of \[x=0\] and \[y=0\] in equation one by one, we get...
Solution: The given inequalities are \[3x+2y\le 150\], \[x+4y\le 80\], \[x\le 5\], \[y\ge 0,x\ge 0\] For \[3x+2y\le 150\] Let us put value of \[x=0\] and \[y=0\] in equation one by one, we get...
Solution: The given inequalities are \[4x+3y\le 60\], \[y\ge 2x\], \[x\ge 3\], \[x,y\ge 0\] For \[4x+3y\le 60\], Let us put value of \[x=0\] and \[y=0\] in equation one by one, we get \[y=20\]and...
Solution: The given inequalities are \[x-2y\le 3\], \[3x+4y\ge 12\], \[x\ge 0\], \[y\ge 1\] For \[x-2y\le 3\] Let us put value of \[x=0\] and \[y=0\] in equation one by one, we get \[y=-1.5\]and...
Solution: The given inequalities are \[2x+y\ge 4\], \[x+y\le 3\], \[2x-3y\le 6\] For \[2x+y\ge 4\], Let us put value of \[x=0\] and \[y=0\] in equation one by one, we get \[y=4\]and \[x=2\] We get...
Solution: The given inequalities are \[3x+4y\le 60\], \[x+3y\le 30\], \[x\ge 0\], \[y\ge 0\] For \[3x+4y\le 60\], Let us put value of \[x=0\] and \[y=0\] in equation one by one, we get \[y=15\]and...
Solution: The given inequalities are \[5x+4y\le 20\], \[x\ge 1\], \[y\ge 2\] For \[5x+4y\le 20\], Let us put value of \[x=0\] and \[y=0\] in equation one by one, we get \[y=5\]and \[x=4\] We get the...
Solution: The given inequalities are \[x+y\le 9\], \[y>x\] , \[x\ge 0\] For \[x+y\le 9\], Let us put value of \[x=0\] and \[y=0\] in equation one by one, we get \[y=9\] and \[x=9\] We get the...
Solution: The given inequalities are \[2x+y\ge 8\], \[x+2y\ge 10\] For \[2x+y\ge 8\] Let us put value of \[x=0\]and \[y=0\]in equation one by one, we get \[y=8\]and \[x=4\] We get the required...
Solution: The given inequalities are \[x+y\le 6\], \[x+y\ge 4\] For \[x+y\le 6\], Let us put the value of \[x=0\] and \[y=0\] in equation one by one, we get \[y=6\] and \[x=6\] We got the points as...
Solution: The given inequalities are \[2x-y>1\], \[x-2y<-1\] \[2x-y>1\]……………… (i) Let us put value of \[x=0\] and \[y=0\]in equation one by one, we get \[y=-1\]and \[x=0.5\] We got the...
Solution: The given inequalities are \[x+y\ge 4,2x-y<0\] For \[x+y\ge 4\] Let us put the value of \[x=0\]and \[y=0\]in equation one by one, we get \[y=4\]and \[x=4\] Therefore, we got points as...
Solution: The given inequalities are \[2x+y\ge 6\]…………… (i) \[3x+4y\le 12\]……………. (ii) For \[2x+y\ge 6\] Take value of \[x=0\] and \[y=0\] in equation one by one, we get value of \[y=6\]and \[x=3\]...
Solution: Given inequalities \[3x+2y\le 12,x\ge 1,y\ge 2\] Let us solve values of x and y by putting \[x=0\]and \[y=0\]one by one we get, \[y=6\]and \[x=4\] We get the points as \[(0,6)\] and...
Solution: The given inequalities are \[x\ge 3\] ……… (i) \[y\ge 2\] …………… (ii) so \[x\ge 3\] that means for any value of y the equation will be unaffected and similarly for \[y\ge 2\], for any value...
Solution: The given inequality is \[x>-3\] Let us draw a dotted line \[x=-3\] in the graph (∵\[x=-3\] is excluded in the given question) Let us Consider \[x>-3\] A point \[(0,0)\] is selected,...
Solution: The given inequality is \[y<-2\] Let us draw a dotted line \[y=-2\] in the graph (∵ \[y=-2\] is excluded in the given question) Let us Consider \[y<-2\] A point \[(0,0)\] is...
Solution: The given inequality is \[y-5x<30\] Let us draw a dotted line \[3y-5x=30\] in the graph (∵\[3y-5x=30\] is excluded in the given question) Let us Consider \[3y-5x<30\] A point...
Solution: The given inequality is \[-3x+2y\ge -6\] Let us draw a solid line \[-3x+2y=-6\] in the graph (∵\[-3x+2y=-6\] is included in the given question) Let us consider \[-3x+2y\ge -6\] A point...
Solution: The given inequality is \[2x-3y>6\] Let us draw a dotted line \[2x-3y=6\] in the graph (∵\[2x-3y=6\] is excluded in the given question) Let us Consider \[2x-3y>6\] A point...
Solution: The given inequality is \[x-y\le 2\] Let us draw a solid line \[x-y=2\] in the graph (∵ \[x-y=2\] is included in the given question) Let us Consider \[x-y\le 2\] A point \[(0,0)\] is...
Solution: The given inequality is \[y+8\ge 2x\] Let us draw a solid line \[y+8=2x\] in the graph (∵\[y+8=2x\] is included in the given question) Let us Consider \[y+8\ge 2x\] A point \[(0,0)\]...
Solution: The inequality given is \[3x+4y\le 12\] Let us draw a solid line \[3x+4y=12\] in the graph (∵\[3x+4y=12\] is included in the given question) Let us consider \[3x+4y\le 12\] A point...
Solution: The given inequality is \[2x+y\ge 6\] Let us draw a solid line \[2x+y=6\] in the graph (∵\[2x+y=6\] is included in the given question) Now take \[2x+y\ge 6\] A point \[(0,0)\] is selected...
Solution: Given inequality \[x+y<5\] Now consider, X \[0\] \[5\] y \[5\] \[0\] Let us draw a dotted line \[x+y=5\] in the graph (∵ \[x+y=5\] is excluded in the given question) Let us take...
Solution: Let us consider the length of the shortest piece be x cm From the question, length of the second piece = \[(x+3)\]cm Given, length of third piece is to be twice as long as the shortest =...
Solution: Let us consider the length of the shortest side of the triangle be x cm From the question, length of the longest side = \[3x\] cm Given, third side is \[2\] cm shorter than the longest...
Solution: Let us consider x be the smaller of the two consecutive even positive integers Then, the other integer = \[x+2\] According to the question that, both the integers are larger than \[5\]...
Solution: Let us consider x be the smaller of the two consecutive odd positive integers Then the other integer = \[x+2\] It is given that both the integers are smaller than \[10\] i.e.,...
Solution: Let us consider Sunita scored x marks in her fifth examination From the given question in order to receive A grade in the course she must have to obtain average \[90\] marks or more in...
Solution: Let us consider x be the marks obtained by Ravi in his third unit test Given that, the entire students should have an average of at least \[60\] marks The inequality based on given...
Solution: The given inequality is \[\frac{x}{2}\ge \frac{(5x-2)}{3}-\frac{(7x-3)}{5}\] After solving we get, \[\frac{x}{2}\ge \frac{5(5x-2)-3(7x-3)}{15}\] Now compute the inequality we get,...
Solution: The inequality given is, \[3(1-x)<2(x+4)\] After Solving the given inequality, we get = \[3-3x<2x+8\] Now after rearranging we get, = \[3-8<2x+3x\] = \[-5<5x\] Divide \[5\] on...
Solution: From the question it is given that, \[5x-3\ge 3x-5\] After solving and rearranging the inequality we get, = \[5x-3x\ge +3-5\] After simplifying we get, \[2x\ge -2\] Divide by 2on both...
Solution: From the question it is given that, \[3x-2<2x+1\] By solving the inequality, we get \[3x-2<2x+1\] = \[3x-2x<2+1\] = \[x<3\] The graph of \[3x-2<2x+1\]is represented below...
Solution: To find inequality of \[\frac{(2x-1)}{3}\ge \frac{(3x-2)}{4}-\frac{(2-x)}{5}\] After rearranging we get, \[\frac{(2x-1)}{3}\ge \frac{15x-10-8+4x}{20}\] = \[\frac{(2x-1)}{3}\ge...
Solution: From the question it is given that \[\frac{x}{4}<\frac{(5x-2)}{3}-\frac{(7x-3)}{5}\] = \[\frac{x}{4}<\frac{5(5x-2)-3(7x-3)}{15}\] After simplifying we get,...
Solution: From the question it is given that, \[37-(3x+5)\ge 9x-8(x-3)\] After simplifying we get = \[37-3x-5\ge 9x-8x+24\] = \[32-3x\ge x+24\] After rearranging we get = \[32-24\ge x+3x\] = \[8\ge...
Solution: From the question it is given that, \[2(2x+3)-10<6(x-2)\] After multiplying we get \[4x+6-10<6x-12\] After simplifying we get \[-4+12<6x-4x\] \[8<2x\] \[4<x\] The solutions...
Solution: From the question it is given that \[\frac{1}{2}\left( \frac{3x}{5}+4 \right)\ge \frac{1}{3}(x-6)\] After cross-multiplying the denominators, we get \[3\left( \frac{3x}{5}+4 \right)\ge...
Solution: From the question it is given that \[\frac{3(x-2)}{5}\le \frac{5(2-x)}{3}\] After cross – multiplying the denominators, we get \[9(x-2)\le 25(2-x)\] \[9x-18\le 50-25x\] Add \[25x\] both...
Solution: From the question it is given that \[\frac{x}{3}>\frac{x}{2}+1\] After rearranging and taking LCM we get, \[\left( \frac{2x-3x}{6} \right)>1\] \[-x/6>1\] \[-x>6\] \[x<-6\]...
Solution: From the question it is given that, \[x+\frac{x}{2}+\frac{x}{3}<11\] Take x as common we get, \[x\left( 1+\frac{1}{2}+\frac{1}{3} \right)<11\] Take LCM we get \[x\left(...
Solution: From the question it is given that, \[3(2-x)\ge 2(1-x)\] After multiplying we get \[6-3x\ge 2-2x\] Adding \[2x\] to both the sides, \[6-3x+2x\ge 2-2x+2x\] \[6-x\ge 2\] Now, subtracting...
Solution: From the question it is given that, \[3(x-1)\le 2(x-3)\] Multiply above inequality can be written as \[3x-3\le 2x-6\] Add \[3\]to both the sides, we get \[3x-3+3\le 2x-6+3\] \[3x\le 2x-3\]...
Solution: From the question it is given that, \[3x-7>5x-1\] Add \[7\] to both the sides, we get \[3x>5x+6\] Again, subtract \[5x\] from both the sides, \[-2x>6\] Divide both sides by...
Solution: From the question it is given that, \[4x+3<5x+7\] subtract \[7\] from both the sides, we get \[4x-4<5x\] Again subtract \[4x\] from both the sides, \[4x-4-4x<5x-4x\] \[x>-4\]...
Solution: (i) From the question \[3x+8>2\] Subtract \[8\]from both sides we get, \[3x>-6\] Divide both sides with \[3\]we get, \[x>-2\] If x is an integer, then the integer number greater...
Solution: (i) According to the given information it is given that \[5x-3<7\] Add \[3\]on both side we get, \[5x-3+3<7+3\] After adding divide both sides by 5 we get, \[5x/5<10/5\]...
Solution: (i) According to the given information \[-12x>30\] Let us divide the inequality by \[-12\] on both sides we get, \[x<-5/2\] If x is a natural integer there is no natural number less...
Solution: (i)According to the information it is given that \[24x<100\] Let us divide the inequality by \[24\] then we get \[x<25/6\] For x is a natural number By inequality natural number less...