solve and graph the inequality using interval notation
(x-4)/(x+8)<0
show work
The inequality holds wherever the numerator and denominator have opposite sign.
Figure out where that happens. x=2 is in the domain of validity. So is x=0
We don't do graphs, but if you draw you own you will get a better understanding of how to do problems like this.
To solve the inequality (x-4)/(x+8) < 0, we need to find the values of x for which the inequality is true.
Step 1: Determine the critical values and the intervals on the number line.
We start by looking for the critical values, which are the values that make the expression (x-4)/(x+8) equal to zero or undefined. In this case, the expression is undefined when x = -8 because it would result in division by zero.
Next, we find the values of x that make the expression equal to zero. In this case, it occurs when (x-4) = 0, leading to x = 4.
Now we have two critical values: -8 and 4. So we mark these values on the number line.
-8 4
Step 2: Test the intervals.
Now we need to test the intervals between the critical values to determine if the expression (x-4)/(x+8) is positive or negative. We can choose any value within each interval to evaluate the inequality.
For the interval (-∞, -8), we can choose x = -9. Plugging this value into the expression, we get:
(-9-4)/(-9+8) = -13/-1 = 13
Since 13 is positive (>0), the expression is positive in this interval.
For the interval (-8, 4), we can choose x = 0. Plugging this value into the expression, we get:
(0-4)/(0+8) = -4/8 = -1/2
Since -1/2 is negative (<0), the expression is negative in this interval.
For the interval (4, ∞), we can choose x = 5. Plugging this value into the expression, we get:
(5-4)/(5+8) = 1/13
Since 1/13 is positive (>0), the expression is positive in this interval.
Step 3: Determine the solution.
Now that we know the sign of the expression for each interval, we can determine the solution to the inequality.
The expression (x-4)/(x+8) is less than zero when the sign changes from positive to negative, which occurs in the interval (-8, 4).
We can represent the solution using interval notation as (-8, 4).
Step 4: Graph the solution.
To graph the solution on a number line, we plot the critical values and shade the corresponding interval. In this case, we shade the interval (-8, 4). The critical values themselves are not included in the solution because the inequality is strict (<) rather than inclusive (≤ or ≥).
-8 4
────●───●──────
The shaded region represents the solution to the inequality (x-4)/(x+8) < 0.
And that's the complete solution to the inequality.