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) ¡Ü 0
James, I gave you a detailed solution yesterday for the same question on the assumption that your symbol was supposed to be ≥
You later told me that it was to be ≤
So why don't you just switch all the conditions
e.g. all the domain that worked before, now will not
and the domain that did not work, now will
you could also follow my Wolfram link and change the ≥ to ≤ , which will reflect the graph in the x-axis.
I am sure you can do that.
http://www.jiskha.com/display.cgi?id=1396018412
I can't make out the inequality sign, but that really makes no difference to the solution
y = (x-4)(x+2)/(x-3)(x+5)
You know there are roots at -2 and 4, and asymptotes at -5 and 3.
So, the graph will cross the x-axis at -2 and 4.
When x < -5, all factors are negative, so y > 0 : --/-- = +
for -5 < x < -2, --/-+ = -
for -2 < x < 3, -+/-+ = +
for 3 < x < 4, -+/++ = -
for x > 4, ++/++ = +
To verify this, check the graph at
http://www.wolframalpha.com/input/?i=%28%28x-4%29%28x%2B2%29%29%2F%28%28x-3%29%28x%2B5%29%29+for+x+%3D+-7..5
To solve the rational inequality (x-4)(x+2)/(x-3)(x+5) ≤ 0, we need to follow these steps:
Step 1: Find the critical points.
We look for the values of x where the numerator and denominator of the rational expression become zero. In this case, the critical points are x = 4, x = -2, x = 3, and x = -5.
Step 2: Create a number line.
Draw a number line and mark the critical points found in Step 1 on the number line.
-5 -2 3 4
|--------|--------|--------|
Step 3: Choose test points.
Select test points from each interval created by the critical points.
For the first interval (-∞, -5), we can choose x = -6.
For the second interval (-5, -2), we can choose x = -4.
For the third interval (-2, 3), we can choose x = 0.
For the fourth interval (3, 4), we can choose x = 3.5.
For the fifth interval (4, ∞), we can choose x = 5.
Step 4: Evaluate the test points.
Substitute the test points from Step 3 into the rational expression and determine the sign of the expression.
For x = -6, the expression is negative.
For x = -4, the expression is positive.
For x = 0, the expression is positive.
For x = 3.5, the expression is negative.
For x = 5, the expression is positive.
Step 5: Analyze the signs and write the solution.
Based on the sign analysis, we can determine the solution to the inequality as follows:
(-∞, -5) ∪ (3, 4]
The solution is represented using interval notation.
Additionally, we can also plot the graph of the rational expression to visually understand the solution.