Math, asked by aggarwalansh10, 1 day ago

Solve the inequality graphically x + 2y ≤10, x + y ≥ 1, x - y ≤0 , x ≥ 0 ,y ≥ 0.

Answers

Answered by mathdude500

$\large\underline{\sf{Solution-}}$

Given inequations are

$\rm :\longmapsto\:x + 2y \leqslant 10$

$\rm :\longmapsto\:x + y \geqslant 1$

$\rm :\longmapsto\:x - y \leqslant 0$

$\rm :\longmapsto\:x \geqslant 0$

$\rm :\longmapsto\:y \geqslant 0$

Consider,

$\rm :\longmapsto\:x + 2y \leqslant 10$

Let we plot the line x + 2y = 10

Hᴇɴᴄᴇ,

➢ Pair of points of the given equation are shown in the below table.

$\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 0 & \sf 5 \\ \\ \sf 10 & \sf 0 \end{array}} \\ \end{gathered}$

Let we check the origin.

$\red{\rm :\longmapsto\:0 + 0 \leqslant 10}$

$\red{\rm :\longmapsto\:0 \leqslant 10 \: which \: is \: true}$

So, the half plane is shaded towards the (0, 0).

Now, Consider

$\rm :\longmapsto\:x + y \geqslant 1$

Now, we plot the line x + y = 1

Hᴇɴᴄᴇ,

➢ Pair of points of the given equation are shown in the below table.

$\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 0 & \sf 1 \\ \\ \sf 1 & \sf 0 \end{array}} \\ \end{gathered}$

Let we check the origin.

$\rm :\longmapsto\:0 + 0 \geqslant 1$

$\rm :\longmapsto\:0 \geqslant 1 \: which \: is \: false$

So, the half plane is shaded away from the origin.

Now, Consider

$\rm :\longmapsto\:x - y \leqslant 0$

Let we plot the line x - y = 0

Hᴇɴᴄᴇ,

➢ Pair of points of the given equation are shown in the below table.

$\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 0 & \sf 0 \\ \\ \sf 1 & \sf 1 \end{array}} \\ \end{gathered}$