Solve the following rational inequality. Write the solution in interval notation. Show the number line and breaking points or graph of the polynomial with the part of the graph that is the solution (x-4)(x+2)/(x-3)(x+5) ¡Ü0
I have a strong feeling you meant
(x-4)(x+2)/( (x-3)(x+5) ) ≥ 0
I good and fast way is to look at the graph of
y = (x-4)( (x+2)/(x-3)(x+5) )
http://www.wolframalpha.com/input/?i=y+%3D+%28x-4%29%28x%2B2%29%2F%28%28x-3%29%28x%2B5%29%29
It shows the graph to be above the x-axis , ( > 0 ) , for x < -5, between -2 and 3 and then again for x > 4
However we must exclude x = 3 and x = -5, since we have vertical asymtotes at these two values
so if your symbol is to be ≥ , then
-2 ≤ x < 3 OR x ≥ 4
another way to algebraically analyse it .
The "critical " values of x are -5, -2, 3, and 4
splitting up the x-axis into 5 sections.
I will then arbitrarily pick any value in each region and calculate using only the ± signs of the factors. We don't really care what the actual answer is, we only care about whether it is + (above) or - (below)
1. for any x < -5
let x = -6, I get (-)(-)/((-)(-) > 0 , that's good
2. for -5<x<-2
let x = -4, I get (-)(-2)/((-)(+)) < 0 , no good
3. for -2 < x < 3
let x = 0 , I get (-)(+)/((-)(+)) > 0 , that's good
4. for 3 < x < 4
let x = 3.5, I get (-)(+)/((+)(+) < 0 , no good
5. for x > 4
let x = 10 , I get (+)(+)/((+)(+)) > 0 , that's good again
so we have:
x < -5 OR -2≤ x < 3 OR x ≥ 4
less than or equal to
To solve the given rational inequality (x-4)(x+2)/(x-3)(x+5) ≤ 0, we first need to find the critical points or breaking points.
The breaking points occur when the numerator or denominator of the rational expression equals zero. So let's set each factor equal to zero and solve for x:
x - 4 = 0 => x = 4
x + 2 = 0 => x = -2
x - 3 = 0 => x = 3
x + 5 = 0 => x = -5
The breaking points occur at x = -5, -2, 3, and 4.
We need to test each interval to find out when the rational expression is less than or equal to zero. We'll choose a test point from each interval and evaluate the expression.
Testing the interval (-∞, -5):
Let's choose x = -6 as the test point. Evaluating the expression at x = -6:
((-6 - 4)(-6 + 2))/((-6 - 3)(-6 + 5)) = (-10 * -4)/(-9 * -1) = 40/9
Since 40/9 > 0, the expression is positive in the interval (-∞, -5).
Testing the interval (-5, -2):
Let's choose x = -3 as the test point. Evaluating the expression at x = -3:
((-3 - 4)(-3 + 2))/((-3 - 3)(-3 + 5)) = (-7 * -1)/(-6 * 2) = 7/12
Since 7/12 > 0, the expression is positive in the interval (-5, -2).
Testing the interval (-2, 3):
Let's choose x = 0 as the test point. Evaluating the expression at x = 0:
((0 - 4)(0 + 2))/((0 - 3)(0 + 5)) = (-4 * 2)/(-3 * 5) = 8/(-15) = -8/15
Since -8/15 < 0, the expression is negative in the interval (-2, 3).
Testing the interval (3, 4):
Let's choose x = 4 as the test point. Evaluating the expression at x = 4:
((4 - 4)(4 + 2))/((4 - 3)(4 + 5)) = (0 * 6)/(1 * 9) = 0
Since 0 = 0, the expression is zero in the interval (3, 4).
Testing the interval (4, ∞):
Let's choose x = 5 as the test point. Evaluating the expression at x = 5:
((5 - 4)(5 + 2))/((5 - 3)(5 + 5)) = (1 * 7)/(2 * 10) = 7/20
Since 7/20 > 0, the expression is positive in the interval (4, ∞).
Now, let's summarize the signs and intervals:
(-∞, -5) : Positive
(-5, -2) : Positive
(-2, 3) : Negative
(3, 4) : Zero
(4, ∞) : Positive
From the above, we can see that the rational expression is less than or equal to zero (-∞) at x = -5 and x = 4 only.
The solution to the rational inequality is x ∈ (-∞, -5] ∪ [4, ∞).
Here is the number line representation of the solution:
---o----o---------o-----------o----
-5 -2 3 4
To solve the rational inequality (x-4)(x+2)/(x-3)(x+5) ≤0, we need to follow a step-by-step process that involves finding the critical points, determining the signs of the factors, and then establishing the solution intervals.
Let's break down the steps:
Step 1: Find the critical points
Critical points are the values of x that make either the numerator (x-4)(x+2) or denominator (x-3)(x+5) equal to zero. In this case, the critical points are x = 4, -2, 3, and -5.
Step 2: Determine the signs of the factors
To determine the signs of the factors, we can use test values and evaluate the signs.
Choose a test value within each of the intervals created by the critical points:
For the interval (-∞, -5), let's choose x = -6.
For the interval (-5, -2), let's choose x = -3.
For the interval (-2, 3), let's choose x = 0.
For the interval (3, 4), let's choose x = 3.5.
For the interval (4, ∞), let's choose x = 5.
Now, plug in each test value into the expression (x-4)(x+2)/(x-3)(x+5):
For x = -6: (-6-4)(-6+2)/(-6-3)(-6+5) = (-10)(-4)/(-9)(-1) = 40/9 (positive)
For x = -3: (-3-4)(-3+2)/(-3-3)(-3+5) = (-7)(-1)/(-6)(2) = 7/6 (positive)
For x = 0: (0-4)(0+2)/(0-3)(0+5) = (-4)(2)/(-3)(5) = 8/15 (negative)
For x = 3.5: (3.5-4)(3.5+2)/(3.5-3)(3.5+5) = (-0.5)(5.5)/(0.5)(8.5) = -2.75/4.25 (negative)
For x = 5: (5-4)(5+2)/(5-3)(5+5) = (1)(7)/(2)(10) = 7/20 (positive)
Step 3: Establish the solution intervals
Based on the signs we found in step 2, we can determine the intervals where the expression is less than or equal to zero (≤0).
(-∞, -5]: The expression is positive/negative/positive. (-∞, -5] is not part of the solution.
[-5, -2): The expression is positive/positive/positive. [-5, -2) is part of the solution.
(-2, 3]: The expression is positive/negative/negative. (-2, 3] is part of the solution.
(3, 4): The expression is negative/negative/positive. (3, 4) is not part of the solution.
[4, ∞): The expression is positive/negative/positive. [4, ∞) is part of the solution.
Step 4: Write the solution in interval notation and present it on a number line
The solution to the rational inequality (x-4)(x+2)/(x-3)(x+5) ≤0 is:
[-5, -2) ∪ (-2, 3]
On a number line, it would look like this:
-6 -5 -4 -3 -2 0 3 4 5
───────────────[───────┬───────)───────────]────────────
(-∞, -5] [-5, -2) (-2, 3] [4, ∞)