solve the inequality and write the answer using interval notation
50/(x+4)+1<4x
50/(x+4) < 4x-1
50 < (x+4)(4x-1)
4x^2 + 15x - 4 > 50
4x^2 + 15x - 54 > 0
(x+6)(4x-9) > 0
critical values are x = -6 , x = 9/4
A quick test for values in each of the 3 domains shows
x < -6 OR x > 9/4
I will let you translate the above into the interval notation you use.
To solve the inequality 50/(x+4) + 1 < 4x, we can follow these steps:
Step 1: Get rid of the fraction by multiplying both sides of the inequality by (x + 4), which is the least common denominator.
50/(x + 4) * (x + 4) + 1 * (x + 4) < 4x * (x + 4)
50 + (x + 4) < 4x^2 + 16x
Step 2: Simplify both sides.
50 + x + 4 < 4x^2 + 16x
x + 54 < 4x^2 + 16x
Step 3: Rearrange the equation by bringing all the terms to one side.
0 < 4x^2 + 16x - x - 54
0 < 4x^2 + 15x - 54
Step 4: Set the equation equal to zero to find the critical points.
4x^2 + 15x - 54 = 0
Step 5: Factor the quadratic equation or use the quadratic formula to solve for x.
(x + 6)(4x - 9) = 0
x + 6 = 0 or 4x - 9 = 0
x = -6 or x = 9/4
Step 6: Plot the critical points on a number line.
Number line: -∞, -6, 9/4, ∞
Step 7: Choose a test value within each interval to determine the sign of inequality.
Choose x = -10, x = 0, x = 2
Step 8: Substitute the test values into the inequality:
For x = -10: 50/(-10 + 4) + 1 < 4(-10) is false.
For x = 0: 50/(0 + 4) + 1 < 4(0) is true.
For x = 2: 50/(2 + 4) + 1 < 4(2) is true.
Step 9: Determine the sign of the inequality in each interval.
(-∞, -6): The inequality is true.
(-6, 9/4): The inequality is false.
(9/4, ∞): The inequality is true.
Step 10: Write the solution in interval notation.
The solution to the inequality 50/(x + 4) + 1 < 4x is (-∞, -6) U (9/4, ∞).