Solve each of the following systems of simultaneous inequations:

3x + 4y _≥ 12, x _≥ 0, y _≥ 1 and 4x + 7y_≤28

Consider the inequation 3x + 4y ≥ 12 :

⇒4y ≥ 12 - 3x

⇒y ≥ 3 -

Consider the equation y = 3 -

Finding points on the coordinate axes:

If x = 0, the y value is 3 i.e, y = 3

⇒ the point on the Y axis is A(0,3)

If y = 0, 0 = 3 -

⇒x = 4

The point on the X axis is B(4,0)

Now consider the inequality y ≥ 3 -

Here we need the y value greater than or equal to y ≥ 3 -

⇒ the required region is above point A.

Therefore the graph of the inequation y ≥ 3 - is fig. 9a

Fig 9a

Consider the inequation 4x + 7y≤28

⇒ 7y≤28 - 4x

⇒y≤4 -

Consider the equation y = 4 -

Finding points on the coordinate axes:

If x = 0, the y value is 4 i.e, y = 4

⇒ the point on the Y axis is C(0,4)

If y = 0, 0 = 4 -

⇒x = 7

The point on the X axis is D(7,0)

Now consider the inequality y≤4 -

Here we need the y value less than or equal to 4 -

⇒ the required region is below point C.

Therefore the graph of the inequation y≤4 - is fig. 9b

Fig 9b

x ≥ 0 is the region right side of Y - axis.

y ≥ 1 is the region above the line y = 1

Combining all the above results in a single graph , we’ll get

The solution of the system of simultaneous inequations is the intersection region of the solutions of the two given inequations.