Q: Solve for x: (x + 3)(x – 4) < 0
A:
Step 1: Find the zeroes
Since the quadratic is already factored, this isn’t too tricky. If it wasn’t factored, you’d have to factor first! (always make sure there is a 0 on one side of the equation / inequality before proceeding).
OK, so what are the zeroes?
x + 3 = 0 or x – 4 = 0
The zeroes are x = -3 or x = 4
So, draw a number line and plot the zeroes on the number line:
Now, you have to test each “interval” that is separated by the “zeroes”. There are three intervals to test.
Interval 1: The numbers to the left of -3 –> in interval notation this is (-infinity, -3)
Interval 2: The numbers between -3 and 4 –> in interval notation this is (-3, 4)
Interval 3: The numbers to the right of 4 –> in interval notation this is (4, infinity)
Step 2: Test each interval
Pick any number on interval 1 and test it into the original inequality. I’ll pick -5:
(x + 3)(x – 4) < 0
(-5 + 3)(-5 – 4) < 0
(-2)(-9) < 0
18 < 0 <— this is false, so numbers on this interval [interval 1] are not part of the solution.
Pick any number on interval 2 and test it into the original inequality. I’ll pick 0:
(x + 3)(x – 4) < 0
(0 + 3)(0 – 4) < 0
(3)(-4) < 0
-12 < 0 <— this is true, so numbers on this interval [interval 2] are a part of the solution.
Pick any number on interval 3 and test it into the original inequality. I’ll pick 5:
(x + 3)(x – 4) < 0
(5 + 3)(5 – 4) < 0
(8)(1) < 0
8 < 0 <— this is false, so numbers on this interval [interval 3] are not part of the solution.
SO: The only interval that “worked” was interval 2. Therefore, the solution is all number between -3 and 4.
In interval notation, we write that like: (-3, 4)
In inequality notation, we write that like: -3 < x < 4