Test a value from each interval to see if it satisfies the original inequality.
Step 1
We will start by solving the related equation.
6x + x^2 ≥ - 9 ⇔ 1x^2+ 6x+ 9 ≥ 0We see above that a= 1, b= 6, and c= 9. Let's substitute these values into the Quadratic Formula to solve the equation.
The solution of the related equation is - 3. Let's plot it on a number line. Since the original is not a strict inequality, the point will be closed.
Step 3
Finally, we must test a value from each interval to see if it satisfies the original inequality. Let's choose a value from the first interval, x ≤ - 3. For simplicity, we will choose x=- 4.
Since x=- 4 produced a true statement, the interval x ≤ - 3 is part of the solution. Similarly, we can test the other interval.
Interval
Test Value
Statement
Is It Part of the Solution?
x ≤ - 3
-4
- 8 ≥ -9 ✓
Yes
x ≥ -3
0
0 ≥ - 9 ✓
Yes
We can now write the solution set and show it on a number line.
x ≤ -3 and x≥ - 3
Since this inequality is not strict, -3 is included in the solution set.
We see above that the solution set is all real numbers.
