Inequalities with two unknowns

Check which of the following points are solutions to the inequality $3x - 2y \le 6$:

• $A(0, 0)$

• $B(4, 1)$

• $C(2, -1)$

• $D(-2, -5)$

Given the inequality $4x - 5y < 20$, state with reasoning whether the points lying exactly on the line $4x - 5y = 20$ form part of the solution region.

Write the linear inequality in two unknowns whose solution region is represented in the following graph. Pay attention to whether the boundary line is solid or dashed.

Graphically represent the solution region of the inequality $x + 2y > 4$ on the Cartesian plane.

You can do it on paper and send a photo with your answer.

Find the system of inequalities that defines the shaded region in the following graph:

Determine whether the points $P(1, 3)$, $Q(-1, 2)$ and $R(4, 0)$ belong to the feasible region defined by the following system of inequalities:

$x + y \le 5$

$2x - y > -1$

Graphically represent the system of inequalities that defines the fourth quadrant of the Cartesian plane (where the x-coordinate is non-negative and the y-coordinate is non-positive).

You can do it on paper and send a photo with your answer.

Graphically represent the feasible region defined by the following system of inequalities:

$2x + y \le 6$

$x - y \ge -1$

$y \ge 0$

You can do it on paper and send a photo with your answer.

To organise a party, soft drink bottles are bought at $2$ € each and bags of snacks at $3$ € each. If the maximum budget is $30$ € and at least one unit of each product must be bought, set up the corresponding system of inequalities and give two possible valid purchase combinations.

In a test with $20$ questions, each correct answer adds $2$ points and each incorrect answer subtracts $1$ point. If a candidate needs at least $15$ points to pass, set up the inequality relating the number of correct answers ($x$) and the number of incorrect answers ($y$). Given that $x \ge 0$, $y \ge 0$ and $x+y \le 20$, graphically represent the region of possible combinations to pass.

You can do it on paper and send a photo with your answer.

A workshop manufactures bicycles and tricycles. Each bicycle requires $2$ wheels and each tricycle $3$ wheels. There are $60$ wheels left in the warehouse. Furthermore, it is known that the number of bicycles manufactured must be at least double the number of tricycles. Set up the system of inequalities that models this situation.

Graphically represent the solution to the quadratic inequality in two unknowns $y \ge x^2 - 4x + 3$.

You can do it on paper and send a photo with your answer.