Solve each inequality. State the solution set using interval notation and
graph it.
x^2 – x – 20 < 0
1. Make sure only 0 is on the right.
This is already the case
2. Factor left side
(x - 5)(x + 4) < 0
1. Find the critical points. These are found by setting the
left hand side = 0 and solving for x
(x-5)(x+4) = 0
x-5=0 gives critical value x=5
x+4=0 gives critical value x = -4
2. Draw a number line and circle the critical values
——————————————o———————————————————————————————————o———————————
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
3. Choose any number left of the leftmost critical point, substitute
it in the factored form of the inequality. If the result is true, shade
that part of the number line, otherwise do not shade it.
Say we choose -5. substitute in:
(x - 5)(x + 4) < 0
(-5 - 5)(-5 + 4) < 0
(-10)(-1) < 0
10 < 0
This is false, so we do not shade the region to the left of -4.
4. Choose any number between the first and second critical points, substitute
it in the factored form of the inequality. If the result is true, shade that
part of the number line, otherwise do not shade it.
Say we choose 0. substitute in:
(x - 5)(x + 4) < 0
(0 - 5)(0 + 4) < 0
(-5)(4) < 0
-20 < 0
This is true, so we shade the region between -4 and +5.
——————————————o===================================o———————————
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
5. Choose any number left of the rightmost critical point, substitute
it in the factored form of the inequality. If the result is true, shade
that part of the number line, otherwise do not shade it.
Say we choose 6. substitute in:
(x - 5)(x + 4) < 0
(6 - 5)(6 + 4) < 0
(1)(10) < 0
10 < 0
This is false, so we do not shade the region to the right of 5.
The interval notation is found by putting the endpoints left to right in
parentheses, with a comma between
Answer: (-4, 5)
Edwin
AnlytcPhil@aol.com