solve the inequality, write answer using interval notation pleas show the work
50/x+4+1<4x
Is there a typo? Why the 4+1?
The problem reads
50 +1 <4x
X+7
so 50divide by x+7
sorry i messed that up
50/x+4 +1<4x
To solve the inequality (50/x + 4 + 1) < 4x, we need to isolate x.
Let's simplify the expression step by step:
1. Start by subtracting 1 from both sides of the inequality:
50/x + 4 < 4x - 1
2. Next, subtract 4 from both sides of the inequality:
50/x < 4x - 5
3. Now, we need to get rid of the fraction (50/x). Multiply both sides of the inequality by x:
x * (50/x) < x * (4x - 5)
This eliminates the fraction:
50 < 4x^2 - 5x
4. Rearrange the equation to solve for x:
4x^2 - 5x - 50 > 0
5. Next, factor the quadratic equation:
(2x - 10)(2x + 5) > 0
6. Now, we can find the critical points by setting each factor to zero and solving for x:
2x - 10 = 0 -> 2x = 10 -> x = 5
2x + 5 = 0 -> 2x = -5 -> x = -5/2 or -2.5
7. Plot these critical points on a number line:
-5/2 -2.5 5
8. Now we need to determine the sign of the expression (2x - 10)(2x + 5) in each interval:
-infinity to -5/2: Choose a value less than -5/2, e.g., -3. Plug it into the expression: (-)(-) = +. So it's positive.
-5/2 to -2.5: Choose a value between -5/2 and -2.5, e.g., -3. Plug it into the expression: (+)(-) = -. So it's negative.
-2.5 to 5: Choose a value greater than -2.5, e.g., 0. Plug it into the expression: (+)(+) = +. So it's positive.
5 to infinity: Choose a value greater than 5, e.g., 6. Plug it into the expression: (+)(+) = +. So it's positive.
9. Based on the sign of the expression, we can conclude that the inequality is satisfied when:
-5/2 < x < -2.5 or x > 5.
Finally, we can write the answer in interval notation:
(-5/2, -2.5) U (5, ∞)