Question

solve the given inequalities; then graph each solution set on separate paper.

1. $x^2 + 7x + 10 > 0$
2. $x^2 + 3x - 18 > 0$

Explanation:

Step1: Factor the quadratic

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

Step2: Find critical points

$x+2=0 \implies x=-2$; $x+5=0 \implies x=-5$

Step3: Test intervals for inequality

• For $x < -5$: $(-)(-) = + > 0$ (valid)
• For $-5 < x < -2$: $(-)(+) = - < 0$ (invalid)
• For $x > -2$: $(+)(+) = + > 0$ (valid)

Step4: Factor the second quadratic

$x^2 + 3x - 18 = (x+6)(x-3)$

Step5: Find critical points

$x+6=0 \implies x=-6$; $x-3=0 \implies x=3$

Step6: Test intervals for inequality

• For $x < -6$: $(-)(-) = + > 0$ (valid)
• For $-6 < x < 3$: $(+)(-) = - < 0$ (invalid)
• For $x > 3$: $(+)(+) = + > 0$ (valid)

Answer:

1. $x < -5$ or $x > -2$
2. $x < -6$ or $x > 3$

(Graphing note: For each solution, plot open circles at the critical points, then shade the valid intervals to the left of the smaller critical point and to the right of the larger critical point.)