solve the inequality and write the answer using interval notation
50/(x+4)+1<4x
To solve the given inequality, let's break it down into steps:
Step 1: Simplify the inequality.
Start by distributing the 50 to (x+4) inside the parenthesis:
50/(x+4) + 1 < 4x becomes 50/(x+4) + 1 < 4x + 4.
Step 2: Get rid of the fractions.
To eliminate the fraction, we can multiply every term in the inequality by the least common multiple (LCM) of the denominators. In this case, the LCM of (x+4) and 1 is (x+4). Multiply each term by (x+4):
(x+4) * 50/(x+4) + (x+4) * 1 < (x+4) * 4x + (x+4) * 4.
This simplifies the inequality to:
50 + (x+4) < 4x(x+4) + 4(x+4).
Step 3: Simplify the inequality further.
50 + x + 4 < 4x^2 + 16x + 4x + 16.
Combine like terms:
x + 54 < 4x^2 + 20x + 16.
Step 4: Rearrange the equation.
Move all the terms to one side of the inequality to set it equal to zero:
4x^2 + 19x + 16 - x - 54 < 0.
Simplify further:
4x^2 + 18x - 38 < 0.
Step 5: Solve for x.
To solve the quadratic inequality, we need to find the values of x that make the quadratic expression less than zero. We can either use factoring, graphing, or the quadratic formula. In this case, let's use factoring.
We need to find two numbers whose product is 4(-38) = -152 and whose sum is 18. The numbers are 20 and -6.
Rewrite the quadratic inequality with factored form:
(2x + 20)(2x - 6) < 0.
Step 6: Determine the sign of the expression.
To determine the sign of the expression, we examine the factors (2x + 20) and (2x - 6). We need to find the intervals where the expression is equal to zero or changes signs.
Setting (2x + 20) = 0, we find x = -10.
Setting (2x - 6) = 0, we find x = 3.
Create a sign chart and test points within each interval:
Test x = -11, (-10, -∞)
(2x + 20)(2x - 6) < 0
(-21)(-26) < 0
546 > 0 (false)
Test x = 0, (-∞, 3)
(2x + 20)(2x - 6) < 0
(20)(-6) < 0
-120 < 0 (true)
Test x = 4, (3, ∞)
(2x + 20)(2x - 6) < 0
(28)(2) < 0
56 < 0 (false)
Step 7: Write the solution using interval notation.
From the sign chart, we can see that the inequality is true when -∞ < x < 3 or 0 < x < 3.
Thus, the solution to the inequality is (-∞, 3) ∪ (0, 3) in interval notation.