Solve algebraically:
4/2x-3 < 1/x+4
Again, watch those needed brackets, assuming you meant:
4/(2x-3) < 1/(x+4)
I see critical values at x = 3/2 and x = -4
try a value x between -4 and 3/2, say x = 0
is 4/(0-3) < 1/(0+4)
is -4/3 < 1/4, YES, so -4 < x < 3/2
try a value x < -4 , say x = -5
is 4/(-10-3) < 1/-1 ?
is -4/13 < -1 ?? , NO
try a value x>3/2, say x = 2
is 4/(8-3) < 1/(2+4) ?
is 4/5 < 1/6 , NO
-4 < x < 3/2
or , look at the graphs of
y = 4/(2x-3) and y = 1/(x+4)
where is the first curve below the second curve?
http://www.wolframalpha.com/input/?i=plot+y+%3D+4%2F(2x-3)+,+y+%3D+1%2F(x%2B4)
Wolfram shows the blue curve below the pink curve between -4 and 1.5
To solve the inequality algebraically, follow these steps:
Step 1: Get rid of the denominators by multiplying both sides of the inequality by the least common denominator (LCD), which is (2x - 3)(x + 4).
(2x - 3)(x + 4) * (4/(2x - 3)) < (2x - 3)(x + 4) * (1/(x + 4))
Simplifying, we have:
4(x + 4) < (2x - 3)(x + 4)
Step 2: Expand and simplify both sides of the inequality.
4x + 16 < 2x^2 + 5x - 12
Step 3: Rearrange the equation to bring all terms to one side to form a quadratic equation.
2x^2 + 5x - 4x - 12 - 16 < 0
2x^2 + x - 28 < 0
Step 4: Solve the quadratic equation.
To solve this quadratic inequality, we can factor or use the quadratic formula. However, this quadratic equation cannot be easily factored, so let's use the quadratic formula.
The quadratic formula is given by x = (-b ± √(b^2 - 4ac))/(2a).
For our equation, a = 2, b = 1, and c = -28.
x = (-(1) ± √((1)^2 - 4(2)(-28))) / (2(2))
x = (-1 ± √(1 + 224)) / 4
x = (-1 ± √225) / 4
x = (-1 ± 15) / 4
x = (-1 + 15) / 4 or x = (-1 - 15) / 4
x = 14 / 4 or x = -16 / 4
x = 7/2 or x = -4
Step 5: Determine the solution.
The solution to the inequality is x < -4 or x > 7/2.