Solve the following inequalities using the number-line method.
a) 3𝑥^2 − 4𝑥 ≤ 15
b) 9 + 9𝑥 − 2𝑥^2 > 0
a) To solve the inequality 3𝑥^2 − 4𝑥 ≤ 15, we first set it equal to zero:
3𝑥^2 − 4𝑥 - 15 ≤ 0
Next, we factor the quadratic equation:
(3𝑥 + 5)(𝑥 - 3) ≤ 0
Now we can find the critical points by setting each factor equal to zero and solving:
3𝑥 + 5 = 0
𝑥 = -5/3
𝑥 - 3 = 0
𝑥 = 3
These critical points divide the number line into three intervals: 𝑥 < -5/3, -5/3 < 𝑥 < 3, and 𝑥 > 3.
To determine the solution, we substitute a test value from each interval into the inequality to see if it is true or false. We can use -2, 0, and 4 as our test values.
For 𝑥 < -5/3, we substitute -2:
(3 * (-2)^2) - (4 * -2) ≤ 15
12 + 8 ≤ 15
20 ≤ 15
This is false, so we exclude 𝑥 < -5/3 from the solution.
For -5/3 < 𝑥 < 3, we substitute 0:
(3 * 0^2) - (4 * 0) ≤ 15
0 ≤ 15
This is true, so we include -5/3 < 𝑥 < 3 in the solution.
For 𝑥 > 3, we substitute 4:
(3 * 4^2) - (4 * 4) ≤ 15
48 - 16 ≤ 15
32 ≤ 15
This is false, so we exclude 𝑥 > 3 from the solution.
Combining the interval that is included in the solution, the solution is -5/3 < 𝑥 < 3.
b) To solve the inequality 9 + 9𝑥 − 2𝑥^2 > 0, we first set it equal to zero:
9 + 9𝑥 - 2𝑥^2 = 0
Next, we factor the quadratic equation:
(3 + 𝑥)(3 - 2𝑥) > 0
Now we can find the critical points by setting each factor equal to zero and solving:
3 + 𝑥 = 0
𝑥 = -3
3 - 2𝑥 = 0
𝑥 = 3/2
These critical points divide the number line into three intervals: 𝑥 < -3, -3 < 𝑥 < 3/2, and 𝑥 > 3/2.
To determine the solution, we substitute a test value from each interval into the inequality to see if it is true or false. We can use -4, -1, and 2 as our test values.
For 𝑥 < -3, we substitute -4:
9 + 9(-4) - 2(-4)^2 > 0
9 - 36 - 2(16) > 0
9 - 36 - 32 > 0
-59 > 0
This is false, so we exclude 𝑥 < -3 from the solution.
For -3 < 𝑥 < 3/2, we substitute -1:
9 + 9(-1) - 2(-1)^2 > 0
9 - 9 - 2(1) > 0
9 - 9 - 2 > 0
-2 > 0
This is false, so we exclude -3 < 𝑥 < 3/2 from the solution.
For 𝑥 > 3/2, we substitute 2:
9 + 9(2) - 2(2)^2 > 0
9 + 18 - 2(4) > 0
9 + 18 - 8 > 0
19 > 0
This is true, so we include 𝑥 > 3/2 in the solution.
Combining the interval that is included in the solution, the solution is 𝑥 > 3/2.