Question

Which ordered pairs make both inequalities true? Check all that apply.

Answer 1

All others did not satisfy the inequality simultaneously except #4 and #6
For #4:
2 < 5(1) + 2 i.e. 2 < 7 (true)
2 >= 1/2(1) + 1 i.e. 2 >= 3/2  (true)
For #6
2 < 5(2) + 2 i.e. 2 < 12 (true)
2 >= 1/2(2) + 1 i.e. 2 >= 2

Answer 2

we have

$y < 5x + 2$ -------> inequality $1$$y \geq \frac{1}{2}x + 1$ -------> inequality $2$

using a graph tool

see the the attached figure N $1$

the solution of the system is the shaded area

$case 1) (-3, 2)$

This point satisfies the inequality $2$ but does not satisfy the inequality $1$.

therefore

it is not a solution of the system

see the attached figure N $2$

inequality $1$

$(-3,2)\\ x=-3\\ y=2$

$2 < 5*(-3) + 2$

$2 < -13$ -----> is not true

inequality $2$

$(-3,2)\\ x=-3\\ y=2$

$2 \geq \frac{1}{2}*(-3) + 1$

$2 \geq -\frac{1}{2}$ -----> is true

$case 2) (-1, 3)$

This point satisfies the inequality $2$ but does not satisfy the inequality $1$.

therefore

it is not a solution of the system

see the attached figure N $2$

inequality $1$

$(-1, 3)\\ x=-1\\ y=3$

$3 < 5*(-1) + 2$

$3 < -3$ -----> is not true

inequality $2$

$(-1, 3)\\ x=-1\\ y=3$

$3 \geq \frac{1}{2}*(-1) + 1$

$3 \geq \frac{1}{2}$ -----> is true

$case 3) (0, 2)$

This point satisfies the inequality $2$ but does not satisfy the inequality $1$.

therefore

it is not a solution of the system

see the attached figure N $2$

inequality $1$

$(0, 2)\\ x=0\\ y=2$

$2 < 5*(0) + 2$

$2 < 2$ -----> is not true

inequality $2$

$(0, 2)\\ x=0\\ y=2$

$2 \geq \frac{1}{2}*(0) + 1$

$2 \geq 1$ -----> is true

$case 4) (1, 2)$

This point satisfies the inequality $2$ and satisfies the inequality $1$.

therefore

it is a solution of the system

see the attached figure N $2$

inequality $1$

$(1, 2)\\ x=1\\ y=2$

$2 < 5*(1) + 2$

$2 < 7$ -----> is true

inequality $2$

$(1, 2)\\ x=1\\ y=2$

$2 \geq \frac{1}{2}*(1) + 1$

$2 \geq \frac{3}{2}$ -----> is true

$case 5) (2,-1)$

This point satisfies the inequality $1$ but does not satisfy the inequality $2$.

therefore

it not a solution of the system

see the attached figure N $2$

inequality $1$

$(2, -1)\\ x=2\\ y=-1$

$-1 < 5*(2) + 2$

$-1 < 12$ -----> is true

inequality $2$

$(2, -1)\\ x=2\\ y=-1$

$-1 \geq \frac{1}{2}*(2) + 1$

$-1 \geq 2$ -----> is not true

$case 6) (2, 2)$

This point satisfies the inequality $1$ and satisfies the inequality $2$.

therefore

it is a solution of the system

see the attached figure N $2$

inequality $1$

$(2, 2)\\ x=2\\ y=2$

$2 < 5*(2) + 2$

$2 < 12$ -----> is true

inequality $2$

$(2, 2)\\ x=2\\ y=2$

$2 \geq \frac{1}{2}*(2) + 1$

$2 \geq 2$ -----> is true

therefore

the answer is

$case 4) (1, 2)\\ case 6) (2, 2)$