Determine whether the statement is true or false.Justify your answer, the solution set of the inequality $\frac { 7 } { 2 } x ^ { 2 } + 7 x + 14 \geq 0$ is the entire set of real numbers.

A)False,the y -values are less than zero for all values of x.
B)True,the y -values are greater than zero for all values of x.

Question

Determine whether the statement is true or false.Justify your answer, the solution set of the inequality $\frac { 7 } { 2 } x ^ { 2 } + 7 x + 14 \geq 0$ is the entire set of real numbers.
​
A)False,the y -values are less than zero for all values of x.
B)True,the y -values are greater than zero for all values of x.

Quizplus · Accepted Answer

The given inequality is a quadratic equation in the form $ax^2 + bx + c \geq 0$, where $a = \frac{7}{2}$, $b = 7$, and $c = 14$. Since $a > 0$, the parabola opens upwards. The discriminant of the quadratic equation is $b^2 - 4ac$, which in this case is $7^2 - 4(\frac{7}{2})(14) = 49 - 196 = -147$. Since the discriminant is negative, the quadratic equation has no real roots and does not cross the x-axis. Therefore, the entire parabola lies above the x-axis, meaning the y-values (or the values of the quadratic expression) are always greater than zero for all real numbers $x$.

Determine Whether the Statement Is True or False 72x2+7x+14≥0\frac { 7 } { 2 } x ^ { 2 } + 7 x + 14 \geq 027​x2+7x+14≥0

Determine Whether the Statement Is True or False $\frac { 7 } { 2 } x ^ { 2 } + 7 x + 14 \geq 0$