I've posted this question before but I forgot to say that I was dealing with polynomial inequalities (not sure if that makes a difference)
The question is: Solve the following polynomial inequalities. (9 marks)
4x - 5 ≤ 2(x - 7)
x^3 - 5x^2 + 2x ≥ -8
2(x^3 - 2x^2 + 3) < x(x - 1)(x + 1)
The answer Reiny previously gave me is the following and I just wanted to double check it:
1.
4x - 5 ≤ 2x - 14
-x ≤ -9
x ≥ 9
2.
x^3 - 5x^2 + 2x + 8 ≥ 0
(x-4)(x-2)(x+1) ≥ 0
critical values are x = 4,2, and -1
the graph of f(x) = (x-4)(x-2)(x+1)
lies above the x-axis between -1 and 2, and values > 4
so -1 ≤ x ≤ 2 OR x ≥ 4
3.
2x^3 - 4x^2 + 6 < x^3 - x
x^3 - 4x^2 + x + 6 < 0
This expression also factors,
hint: (x+1) is a factor
I will let you finish it, let me know what you got
To solve polynomial inequalities, you can follow these steps:
1. Simplify the expressions on both sides of the inequality.
2. Set the expression equal to zero and factorize it if possible.
3. Determine the critical values by setting each factor equal to zero and solving for x.
4. Plot the critical values on a number line and choose test points between the critical values.
5. Substitute these test points into the original inequality to determine the sign of the expression within each interval.
6. Write the solution using interval notation or set notation based on the signs of the expression.
Let's go through the solutions for each polynomial inequality:
1. 4x - 5 ≤ 2(x - 7)
Start by distributing 2 on the right side: 4x - 5 ≤ 2x - 14
Combine like terms: 2x ≤ -9
Divide by 2: x ≥ -9/2
So the solution is x ≥ -9/2.
2. x^3 - 5x^2 + 2x ≥ -8
First, set the expression equal to zero: x^3 - 5x^2 + 2x + 8 ≥ 0
Factorize the expression: (x - 4)(x - 2)(x + 1) ≥ 0
Find the critical values by setting each factor equal to zero: x = 4, 2, -1
Plot these critical values on a number line:
-Infinity (-1) | (2) | (4) Infinity
Test a value in each interval to determine the sign of the expression:
- Substitute x = -2: (-)(-)(-) ≥ 0, which is true.
- Substitute x = 3: (+)(+)(-) ≥ 0, which is false.
- Substitute x = 5: (+)(+)(+) ≥ 0, which is true.
Based on the signs, the solution is -1 ≤ x ≤ 2 or x ≥ 4.
3. 2(x^3 - 2x^2 + 3) < x(x - 1)(x + 1)
Start by simplifying both sides of the inequality: 2x^3 - 4x^2 + 6 < x^3 - x
Combine like terms: x^3 - 4x^2 + x + 6 < 0
Factorize the expression: (x + 1)(x - 2)(x^2 - x + 6) < 0
The factor (x + 1) tells us that x = -1 is a critical value.
To find the critical values from the quadratic, solve for x where the discriminant (b^2 - 4ac) is greater than 0:
Discriminant = (-1)^2 - 4*1*6 = 1 - 24 = -23, which is less than 0.
Since the quadratic doesn't have real solutions, x = -1 is the only critical value.
Plotting the critical value on a number line: (-∞) -1 (Infinity)
Test a value in each interval:
- Substitute x = -2: (+)(-)(+) < 0, which is true.
- Substitute x = 0: (+)(-)(+) < 0, which is true.
Based on the signs, the solution is -∞ < x < -1.