How do you solve and write the following in interval notation: #x^2 + 7x + 10 0#?

Question

How do you solve and write the following in interval notation: #x^2 + 7x + 10 >0#?

Julia Adams · Accepted Answer

The solution is #x in (-oo,-5) uu (-2,+oo)#

Let's divide the inequality by two.

#x^2+7x+10=(x+2)(x+5)#

Thus,

#(x+2)(x+5)>0#

Let #f(x)=(x+2)(x+5)#

We create a sign chart.

#color(white)(aaaa)##x##color(white)(aaaaa)##-oo##color(white)(aaaa)##-5##color(white)(aaaa)##-2##color(white)(aaaa)##+oo#

#color(white)(aaaa)##x+5##color(white)(aaaaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##+#

#color(white)(aaaa)##x+2##color(white)(aaaaaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##+#

#color(white)(aaaa)##f(x)##color(white)(aaaaaaaa)##+##color(white)(aaaa)##-##color(white)(aaaa)##+#

Consequently,

#f(x)>0# when # x in (-oo,-5) uu (-2,+oo)#

Eva Abramson · Answer

undefined To solve $x^2 + 7x + 10 > 0$ and write the solution in interval notation, follow these steps:

1. Factor the quadratic expression: $(x + 5)(x + 2) > 0$.
2. Find the critical points by setting each factor equal to zero: $x + 5 = 0$ and $x + 2 = 0$. Solving these equations yields $x = -5$ and $x = -2$.
3. Plot these critical points on a number line.
4. Test intervals between and outside of the critical points using test points to determine where the expression is greater than zero.
5. Write the solution in interval notation based on the intervals where the expression is greater than zero.

The solution in interval notation is $(-\infty, -5) \cup (-2, \infty)$.

How do you solve and write the following in interval notation: #x^2 + 7x + 10 &gt;0#?

How do you solve and write the following in interval notation: #x^2 + 7x + 10 >0#?