Solve the given inequalities
Question: Solve the given inequalities $2 x+y \geq 4, x+y \leq 3,2 x-3 y \leq 6$ graphically in two-dimensional plane. Solution: The graphical representation of $2 x+y \geq 4, x+y \leq 3,2 x-3 y \leq 6$ is given by common region in the figure below. $2 x+y \geq 4 \ldots \ldots$ (1) $x+y \leq 3 \ldots \ldots$ (2) $2 x-3 y \leq 6 \ldots \ldots$ (3) Inequality (1) represents the region above line $2 x+y=4$ (including the line $2 x+y=4$ ). Inequality (2) represents the region below line $x+y=3$ (inc...
Read More →The volume of water in a tank is twice
Question: The volume of water in a tank is twice of that in the other. If we draw out 25 litres from the first and add it to the other, the volumes of the water in each tank will be the same. Find the volumes of water in each tank. Solution: Let volume of water in one tank be $x \mathrm{~L}$. Then, volume of the water in another $\tan k=2 x \mathrm{~L}$. According to the question, Volume of the water in first tank $+25=$ Volume of the water in another tank $-25$ $\Rightarrow$ $x+25=2 x-25$ $\Rig...
Read More →Rs 13500 are to be distributed among Salma,
Question: Rs 13500 are to be distributed among Salma, Kiran and Jenifer in such a way that Salma gets Rs 1000 more than Kiran and Jenifer gets Rs 500 more than Kiran. Find the money received by Jenifer. Solution: Let the money received by Kiran be $₹ x$. Then, the money received by Salma $=₹(x+1000)$ and the money received by Jenifer $=₹(x+500)$ According to the question, $x+x+1000+x+500=13500$ $\Rightarrow$ $3 x+1500=13500$ $\Rightarrow$ $3 x=13500-1500$ $\Rightarrow \quad 3 x=12000$ $\Rightarr...
Read More →Solve the given inequalities
Question: Solve the given inequalities $3 x+4 y \leq 60, x+3 y \leq 30, x \geq 0, y \geq 0$ graphically in two - dimensional plane. Solution: The graphical representation of $3 x+4 y \leq 60, x+3 y \leq 30, x \geq 0, y \geq 0$ is given by common region in the figure below. $3 x+4 y \leq 60 \ldots \ldots$ (1) $x+3 y \leq 30 \ldots \ldots$ (2) $x \geq 0 \ldots \ldots$ (3) $y \geq 0 \ldots \ldots$ (4) Inequality (1) represents the region below line $3 x+4 y=60$ (including the line $3 x+4 y=60$ ). I...
Read More →Radha takes some flowers in a basket
Question: Radha takes some flowers in a basket and visits three temples one-by-one. At each temple, she offers one half of the flowers from the basket. If she is left with 3 flowers at the end, then find the number of flowers she had in the beginning. Solution: Let total number of the flowers be $x$. Then, flowers offered in first temple $=\frac{x}{2}$ Flowers offered in second temple $=\frac{x}{4}$ Flowers offered in third temple $=\frac{x}{8}$ Number of flowers left at the end $=3$ According t...
Read More →0.16 (5x-2)=0.4x +7
Question: 0.16 (5x-2)=0.4x +7 Solution: Given, $0.16(5 x-2)=0.4 x+7$ $\Rightarrow$ $0.8 x-0.32=0.4 x+7$ $\Rightarrow$ $0.8 x-0.4 x=0.32+7$ [transposing $0.4 x$ to LHS and $-0.32$ to RHS] $\Rightarrow \quad 0.4 x=7.32$ $\Rightarrow$ $\frac{0.4 x}{0.4}=\frac{7.32}{0.4}$ [dividing both sides by 0.4] $\therefore$ $x=18.3$...
Read More →Solve the given inequalities
Question: Solve the given inequalities $5 x+4 y \leq 20, x \geq 1, y \geq 2$ graphically in two - dimensional plane. Solution: The graphical representation of $5 x+4 y \leq 20, x \geq 1, y \geq 2$ is given by common region in the figure below. $5 x+4 y \leq 20$ . (1) $x \geq 1 \ldots \ldots .(2)$ $y \geq 2 \ldots \ldots .(3)$ Inequality $(1)$ represents the region below line $5 x+4 y=20$ (including the line $5 x+4 y=20$ ). Inequality (2) represents the region in front of line $x=1$ (including th...
Read More →3(5x-2)+2(9x-11)=4(8x-7)-111
Question: 3(5x-2)+2(9x-11)=4(8x-7)-111 Solution: Given, $\quad 3(5 x-7)+2(9 x-11)=4(8 x-7)-111$ $\Rightarrow \quad 15 x-21+18 x-22=32 x-28-111$ $\Rightarrow \quad 33 x-43=32 x-139$ $\Rightarrow$ $33 x-32 x=-139+43$ [transposing $32 x$ to LHS and $-43$ to RHS] $\therefore$ $x=-96$...
Read More →4 (3p + 2) – 5 (6p – 1) = 2 (p – 8) – 6 (7p – 4)
Question: 4 (3p + 2) 5 (6p 1) = 2 (p 8) 6 (7p 4) Solution: Given, $4(3 p+2)-5(6 p-1)=2(p-8)-6(7 p-4)$ $\Rightarrow$ $12 p+8-30 p+5=2 p-16-42 p+24$ $\Rightarrow \quad-18 p+13=-40 p+8$ $\Rightarrow$ $-18 p+40 p=8-13 \quad$ [transposing $-40 p$ to LHS and 13 to RHS ] $\Rightarrow \quad 22 p=-5$ $\Rightarrow \quad \frac{22 p}{22}=\frac{-5}{22}$ [dividing both sides by 22] $\therefore$ $p=\frac{-5}{22}$...
Read More →Solve the given inequalities
Question: Solve the given inequalities $2 x-y1, x-2 y1$ graphically in two-dimensional plane. Solution: The graphical representation of $2 x-y1, x-2 y1$ is given by common region in the figure below. $2 x-y1 \ldots \ldots$ (1) $x-2 y1 \ldots \ldots(2)$ Inequality (1) represents the region below line $2 x-y=1$ (excluding the line $2 x-y=1$ ). Inequality (2) represents the region above line $x-2 y=1$ (excluding the line $x-2 y=1$ ). Therefore,every point in the common shaded region including the p...
Read More →prove that
Question: $m-\frac{m-1}{2}=1-\frac{m-2}{3}$ Solution: Given $m-\frac{m-1}{2}=1-\frac{m-2}{3}$ $\Rightarrow$ $\frac{2 m-(m-1)}{2}=\frac{3-(m-2)}{3}$ $\Rightarrow$ $3(2 m-m+1)=2(3-m+2)$ [by cross multiplication] $\Rightarrow \quad 3(m+1)=2(5-m)$ $\Rightarrow \quad 3 m+3=10-2 m$ $\Rightarrow$ $3 m+2 m=10-3$ [transposing $-2 m$ to LHS and 3 to RHS] $\Rightarrow$ $5 m=7$ $\Rightarrow$ $\frac{5 m}{5}=\frac{7}{5}$ [dividing both sides by 5 ] $\therefore$ $m=\frac{7}{5}$...
Read More →Solve the given inequalities
Question: Solve the given inequalities $x+y \leq 9, yx, x \geq 0$ graphically in two-dimensional plane. Solution: The graphical representation of $x+y \leq 9, yx, x \geq 0$ is given by common region in the figure below. $x+y \leq 9 \ldots \ldots$ (1) $yx \ldots \ldots$ (2) $x \geq 0 \ldots \ldots$ (3) Inequality (1) represents the region below line $x+y=9$ (including the line $x+y=9$ ). Inequality (2) represents the region below line $x=y$ (excluding the line $x=y$ ). Inequality (3) represents t...
Read More →prove that
Question: $\frac{3 t-2}{3}+\frac{2 t+3}{2}=t+\frac{7}{6}$ Solution: Given, $\frac{3 t-2}{3}+\frac{2 t+3}{2}=t+\frac{7}{6}$ $\Rightarrow$ $\frac{2(3 t-2)+3(2 t+3)}{6}=\frac{6 t+7}{6}$ $\Rightarrow$ $6 t-4+6 t+9=6 t+7$ $\Rightarrow$ $12 t+5=6 t+7$ $\Rightarrow$ $12 t-6 t=7-5$ [transposing $6 t$ to LHS and 5 to RHS] $\Rightarrow$ $6 t=2$ $\Rightarrow$ $\frac{6 t}{6}=\frac{2}{6}$ [dividing both sides by 6 ] $\therefore$ $t=\frac{1}{3}$...
Read More →prove that
Question: $\frac{3 t-2}{3}+\frac{2 t+3}{2}=t+\frac{7}{6}$ Solution: Given, $\frac{5 x+1}{2 x}=-\frac{1}{3}$ $\Rightarrow \quad 3(5 x+1)=-2 x$ [by cross-multiplication] $\Rightarrow \quad 15 x+3=-2 x$ $\Rightarrow$ $15 x+2 x=-3$ [transposing $-2 x$ to LHS and 3 to RHS] $\Rightarrow \quad 17 x=-3$ $\Rightarrow$ $\frac{17 x}{17}=\frac{-3}{17}$ [dividing both sides by 17 ] $\therefore$ $x=\frac{-3}{17}$...
Read More →Solve the given inequalities
Question: Solve the given inequalities $2 x+y \geq 6,3 x+4 y \leq 12$ graphically in two - dimensional plane. Solution: The graphical representation of $2 x+y \geq 6,3 x+4 y \leq 12$ is given by common region in the figure below. $2 x+y \geq 6 \ldots \ldots$ (1) $3 x+4 y \leq 12 \ldots \ldots$ (2) Inequality (1) represents the region above line $2 x+y=6$ (including the line $2 x+y=6$ ). Inequality (2) represents the region below line $3 x+4 y=12$ (including the line $3 x+4 y=12$ ). Therefore,eve...
Read More →prove that
Question: $\frac{3 x+2}{2 x-3}=-\frac{3}{4}$ Solution: Given, $\frac{3 x+2}{2 x-3}=-\frac{3}{4}$ $\Rightarrow \quad 4(3 x+2)=-3(2 x-3)$ [by cross-multiplication] $\Rightarrow \quad 12 x+8=-6 x+9$ $\Rightarrow$ $12 x+6 x=9-8$ [transposing - $6 x$ to LHS and 8 to RHS] $\Rightarrow$ $18 x=1$ $\Rightarrow$ $\frac{18 x}{18}=\frac{1}{18}$ [dividing both sides by 18] $\therefore$ $x=\frac{1}{18}$...
Read More →Solve the given inequalities
Question: Solve the given inequalities $x+y \leq 6, x+y \geq 4$ graphically in two-dimensional plane. Solution: The graphical representation of $x+y \leq 6, x+y \geq 4$ is given by common region in the figure below. $x+y \leq 6 \ldots \ldots(1)$ $x+y \geq 4 \ldots \ldots(2)$ Inequality (1) represents the region below line $x+y=6$ (including the line $x+y=6$ ). Inequality (2) represents the region above line $x+y=4$ (including the line $x+y=4$ ). Therefore, every point in the common shaded region...
Read More →Solve the given inequalities
Question: Solve the given inequalities $3 x+2 y \leq 12, x \leq 1, y \geq 2$ graphically in two - dimensional plane. Solution: The graphical representation of $3 x+2 y \leq 12, x \leq 1, y \geq 2$ is given by common region in the figure below. $3 x+2 y \leq 12 \ldots \ldots$ (1) $x \leq 1 \ldots \ldots .(2)$ $y \geq 2 \ldots \ldots$ (3) Inequality $(1)$ represents the region below line $3 x+2 y=12$ (including the line $3 x+2 y=12$ ). Inequality (2) represents the region behind line $x=1$ (includ...
Read More →prove that
Question: $\frac{9-3 y}{1-9 y}=\frac{8}{5}$ Solution: Given, $\frac{9-3 y}{1-9 y}=\frac{8}{5}$ $\Rightarrow$ $5(9-3 y)=8(1-9 y)$ [by cross-multiplication] $\Rightarrow \quad 45-15 y=8-72 y$ $\Rightarrow \quad 72 y-15 y=8-45 \quad$ [transposing $-72 y$ to LHS and 45 to RHS] $\Rightarrow \quad 57 y=-37$ $\Rightarrow$ $\frac{57 y}{57}=\frac{-37}{57}$ [dividing both sides by 57 ] $\therefore$ $y=\frac{-37}{57}$...
Read More →Solve the given inequalities
Question: Solve the given inequalities $x \geq 2 y, y \geq 3$ graphically in two-dimensional plane. Solution: The graphical representation of $x \geq 2 y, y \geq 3$ is given by common region in the figure below. $x \geq 2 y \ldots \ldots(1)$ $y \geq 3 \ldots \ldots(2)$ Inequality (1) represents the region below line $x=2 y$ (including the line $x=2 y$ ). Inequality $(2)$ represents the region above line $y=3($ including the line $y=3)$. Therefore, every point in the common shaded region includin...
Read More →Solve the given inequality
Question: Solve the given inequality $3 x+5 y15$ graphically in two-dimensional plane. Solution: The graphical representation of $3 x+5 y15$ is given by blue dotted line in the figure below. This lines divides $x-y$ plane into two parts. Select a point (not on the line), which lies on one of the two parts, to determine whether the point satisfies the given inequality or not. We select the point as $(0,0)$ It is observed that $0+015$ or $015$ which is true. Therefore, the solution for the given i...
Read More →Solve the given inequality
Question: Solve the given inequality $3 x+2 y6$ graphically in two - dimensional plane. Solution: The graphical representation of $3 x+2 y6$ is given by blue dotted line in the figure below. This lines divides $x-y$ plane into two parts. Select a point (not on the line), which lies on one of the two parts, to determine whether the point satisfies the given inequality or not. We select the point as $(0,0)$ It is observed that $0+06$ or $06$ which is false. Therefore, the solution for the given in...
Read More →0.25(4x-5)=0.75x +8
Question: 0.25(4x-5)=0.75x +8 Solution: Given, $0.25(4 x-5)=0.75 x+8$ $\Rightarrow \quad x-1.25=0.75 x+8$ $\Rightarrow$ $x-0.75 x=1.25+8$ [transposing $0.75 x$ to LHS and $1.25$ to RHS] $\Rightarrow$ $0.25 x=9.25$ $\Rightarrow$ $\frac{0.25 x}{0.25}=\frac{9.25}{0.25}$ [dividing both sides by $0.25$ ] $\therefore$ $x=37$...
Read More →Solve the given inequality
Question: Solve the given inequality $x \geq y-2$ graphically in two - dimensional plane. Solution: The graphical representation of $x \geq y-2$ is given by blue dotted line in the figure below. This lines divides $x-y$ plane into two parts. Select a point (not on the line), which lies on one of the two parts, to determine whether the point satisfies the given inequality or not. We select the point as $(0,0)$ It is observed that $00-2$ or $0-2$ which is false. Therefore, the solution for the giv...
Read More →solve that
Question: $\frac{2 y-3}{4}-\frac{3 y-5}{2}=y+\frac{3}{4}$ Solution: Given, $\frac{2 y-3}{4}-\frac{3 y-5}{2}=y+\frac{3}{4}$ $\Rightarrow$ $\frac{2 y-3-2(3 y-5)}{4}=\frac{4 y+3}{4}$ $\Rightarrow \quad 2 y-3-6 y+10=4 y+3$ $\Rightarrow$ $-4 y+7=4 y+3$ [transposing $4 y$ to LHS and 7 to RHS] $\Rightarrow$ $-4 y-4 y=3-7$ $\Rightarrow$ $-8 y=-4$ $\Rightarrow$ $\frac{-8 y}{-8}=\frac{-4}{-8}$ [dividing both sides by $-8$ ] $\therefore$ $y=\frac{1}{2}$...
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