Question

Select the point that is a solution to the system of inequalities

Answer 1

Hello!

The answer is:

The point that is a solution to the system of inequalities is C(4,2).

Why?

To find the point that is a solution to the system of inequalities, we need to evaluate it into the given inequalities. If the point is a solution to the system of inequalities, both inequalities will be satisfied.

We are given the inequalities:

$y\leq 2x-2$

$y\leq x^{2} -3x$

So, substituting the given points into the given inequalities, we have:

- A. (2,1)

Substituting into the first inequality, we have:

$y\leq 2x-2\\\\1\leq 2*(2)-2\\\\1\leq 4-2\\\\1\leq 2$

Substituting into the second inequality, we have:

$y\leq x^{2} -3x$

$1\leq (2)^{2} -3(2)$

$1\leq (2)^{2} -3(2)$

$1\leq 4 -6$

$1\leq -2$

Therefore, since 1 is not less or equal to -2, the point A(2,1) is not a solution to the system of inequalities.

- B. (-2,-1)

Substituting into the first inequality, we have:

$y\leq 2x-2\\\\-1\leq (-2)*(2)-2\\\\-1\leq -4-2\\\\-1\leq -6$

Substituting into the second inequality, we have:

$y\leq x^{2} -3x$

$-1\leq (-2)^{2} -3(-2)$

$-1\leq 4 +6$

$-1\leq 10$

Therefore, since -1 is not less or equal to -6, the point B(-2,-1) is not a solution to the system of inequalities.

- C. (4,2)

Substituting into the first inequality, we have:

$y\leq 2x-2\\\\2\leq (4)*(2)-2\\\\2\leq 8-2\\\\2\leq 6$

Substituting into the second inequality, we have:

$y\leq x^{2} -3x$

$2\leq (4)^{2} -3(4)$

$2\leq 16 -12$

$2\leq 4$

Therefore, since 2 is less than 6, and 2 is less than 4, the point B(4,2) is a solution to the system of inequalities.

- D(1,3)

Substituting into the first inequality, we have:

$y\leq 2x-2\\\\3\leq (1)*(2)-2\\\\3\leq 2-2\\\\3\leq 0$

Substituting into the second inequality, we have:

$y\leq x^{2} -3x$

$3\leq (1)^{2} -3(1)$

$3\leq 1 -3$

$3\leq -2$

Therefore, since 3 is not less than 0, and 3 is not less than -2, the point D(1.3) is not a solution to the system of inequalities.

Hence, the point that is a solution to the system of inequalities is C(4,2).

Have a nice day!