Solve for x under the assumption that x>0. Enter your answer in interval notion using grouping symbols.
X-48/x<-8
[x-48]/x<-8
x-48<-8x
9x<48
x<48/9
(x-48)/x < -8
You cannot just multiply by x as shown above, because x might be negative, which would then reverse the direction of the inequality.
If x > 0, then
x-48 < -8x
9x < 48
x < 48/9
so, 0 < x < 48/9
If x < 0, then
x-48 > -8x
9x > 48
x > 48/9
But that's out, since we assumed x<0.
If you plot the curve, you will see that a simple "x < 48/9" is not quite true.
OR
(x-48)/x < -8
(x-48)/x + 8 < 0
(x-48 + 8x)/x < 0
(9x - 48)/x < 0
but we are told that x > 0, then
9x-48 < 0
9x < 48
x < 48/9
To solve the inequality x - 48/x < -8, we'll follow these steps:
Step 1: Bring all terms to one side of the inequality:
x - 48/x + 8 < 0
Step 2: Combine the terms:
(x^2 - 48 + 8x)/x < 0
Step 3: Find the critical points where the expression in the numerator becomes zero:
x^2 + 8x - 48 = 0
Factoring or using the quadratic formula, we find:
(x + 12)(x - 4) = 0
So the critical points are x = -12 and x = 4.
Step 4: Create a sign chart to determine the intervals where the inequality is true or false:
--- | --- -12 --- 4 --- [Increase] --- [Decrease] ---
Choose a test point from each interval to determine the sign of the expression:
For x < -12:
Choose x = -15:
((-15)^2 - 48 + 8(-15))/(-15) = (-15 + 180 - 120)/(-15) = 3/(-1) = -3 (negative)
For -12 < x < 4:
Choose x = 0:
(0^2 - 48 + 8(0))/0 = (-48)/0 = undefined
For x > 4:
Choose x = 5:
((5)^2 - 48 + 8(5))/(5)= (25 - 48 + 40)/(5) = 17/5 = 3.4 (positive)
Step 5: Analyzing the sign chart, we observe that the inequality is true for x < -12 or 4 < x.
Step 6: Write the solution in interval notation using grouping symbols:
(-∞,-12) U (4,∞)
Therefore, the solution to the given inequality, under the assumption that x > 0, is (-∞,-12) U (4,∞) in interval notation.