Which of the following ordered pairs is a solution to the system of inequalities given below? x+ylt 4 x^2+ylt 8 A (1,5) B (1,2) C (4,-1) D (-4,-1)

To determine which ordered pair is a solution to the system of inequalities, we need to substitute each pair into the inequalities and check if they satisfy both conditions.<br /><br />Let's start with option A: $(1,5)$<br /><br />Substituting $x=1$ and $y=5$ into the first inequality, we get:<br />$1+5 < 4$<br />$6 < 4$ (False)<br /><br />Since the first inequality is not satisfied, we can eliminate option A.<br /><br />Next, let's try option B: $(1,2)$<br /><br />Substituting $x=1$ and $y=2$ into the first inequality, we get:<br />$1+2 < 4$<br />$3 < 4$ (True)<br /><br />Now, let's substitute $x=1$ and $y=2$ into the second inequality:<br />$1^2 + 2 < 8$<br />$1 + 2 < 8$<br />$3 < 8$ (True)<br /><br />Since both inequalities are satisfied, option B is a solution to the system of inequalities.<br /><br />We can now check options C and D for completeness.<br /><br />Option C: $(4,-1)$<br /><br />Substituting $x=4$ and $y=-1$ into the first inequality, we get:<br />$4+(-1) < 4$<br />$3 < 4$ (True)<br /><br />Substituting $x=4$ and $y=-1$ into the second inequality, we get:<br />$4^2 + (-1) < 8$<br />$16 - 1 < 8$<br />$15 < 8$ (False)<br /><br />Since the second inequality is not satisfied, we can eliminate option C.<br /><br />Option D: $(-4,-1)$<br /><br />Substituting $x=-4$ and $y=-1$ into the first inequality, we get:<br />$-4+(-1) < 4$<br />$-5 < 4$ (True)<br /><br />Substituting $x=-4$ and $y=-1$ into the second inequality, we get:<br />$(-4)^2 + (-1) < 8$<br />$16 - 1 < 8$<br />$15 < 8$ (False)<br /><br />Since the second inequality is not satisfied, we can eliminate option D.<br /><br />Therefore, the correct answer is option B: $(1,2)$.