We are learning to solve Factorable Polynomial Inequalities Algebraically.
Question:
Solve using intervals.
(4x +1)(2-x) is greater than or equal to 0.
Set equal factor to ≥ o
(4x+1)≥0
(2-x)≥0
Now solve for x on each one
To solve the inequality (4x + 1)(2 - x) ≥ 0 algebraically using intervals, you can follow these steps:
Step 1: Find the critical numbers
The critical numbers are the values of x for which the expression (4x + 1)(2 - x) is equal to zero. In this case, we can set each factor equal to zero and solve for x:
4x + 1 = 0 --> 4x = -1 --> x = -1/4
2 - x = 0 --> x = 2
So, the critical numbers for this inequality are x = -1/4 and x = 2.
Step 2: Create intervals
To create intervals, you need to plot the critical numbers on a number line and choose test points within each interval. Since there are two critical numbers, you can divide the number line into three intervals:
Interval 1: x < -1/4
Interval 2: -1/4 < x < 2
Interval 3: x > 2
Step 3: Choose test points
Within each interval, choose a test point that is not a critical number. It is usually convenient to choose integers:
For Interval 1: Choose x = -2
For Interval 2: Choose x = 0
For Interval 3: Choose x = 3
Step 4: Evaluate the inequality
Substitute the test points into the original inequality and determine if it is true or false:
For Interval 1: (4(-2) + 1)(2 - (-2)) = (-7)(4) = -28 < 0 (FALSE)
For Interval 2: (4(0) + 1)(2 - 0) = (1)(2) = 2 > 0 (TRUE)
For Interval 3: (4(3) + 1)(2 - 3) = (13)(-1) = -13 < 0 (FALSE)
Step 5: Identify the solution
From the test point evaluations:
- For Interval 1, the inequality is false.
- For Interval 2, the inequality is true.
- For Interval 3, the inequality is false.
The solution to the inequality (4x + 1)(2 - x) ≥ 0 is x ∈ (-1/4, 2].
This means that x is greater than -1/4 and less than or equal to 2.