4/x+5>1/2x+3
I usually do these the following way:
consider the actual equation 4/(x+5) = 1/(2x+3)
easy to solve and x = -1
also x ≠ -5, -3/2
So we have the following critical parts of the domain for x
x < -5
-5 < x < -3/2
-3/2 < x < -1 , and
x > -1
pick a value in each section
for x < -5, let x = -6 .... is 4/-1 > 1/-9 ?? , NO
-5 < x < -3/2, let x = -3 ... is 4/2 > 1/-3 ?? , YES
-3/2 < x < -1, let x = -1.1 ... is 4/3.9 > 1/.8 ??, NO
finally, x > -1, let x = 10, ..... is 4/15 > 1/23 ?? YES
so we have:
-5 < x < -3/2 OR x > -1
graphing y = 4/(x+5) and y= 1/(2x+3) using DESMOS shows this is correct.
By squaring both sides, sometimes solutions are introduced which do
not satisfy the original relation. That is why all solutions must be verified after squaring.
The value of x = -17/9 does not show up when graphing
well said; always check your answers against the original equations.
Unfettered cleverness will get you in trouble.
Can't beat being unfettered no matter what !
To solve the inequality 4/x + 5 > 1/2x + 3, we will follow these steps:
1. Eliminate the fractions:
To eliminate the fractions, we can multiply both sides of the inequality by the least common denominator (LCD) of the fractions involved. The LCD in this case is 2x, so we multiply both sides of the inequality by 2x:
2x * (4/x) + 2x * 5 > 2x * (1/2x) + 2x * 3
Simplifying the equation, we have:
8 + 10x > 1 + 6x
2. Simplify the equation:
Combine like terms on both sides of the inequality. In this case, we will combine the constants and the terms with x:
10x - 6x > 1 - 8
4x > -7
3. Solve for x:
To solve for x, divide both sides of the inequality by 4:
(4x)/4 > -7/4
x > -7/4
Therefore, the solution to the inequality 4/x + 5 > 1/2x + 3 is x > -7/4, which means x is greater than -7/4.
assuming the usual carelessness with parentheses,
4/(x+5) > 1/(2x+3)
you can work out four separate solutions, choosing which expressions are positive or negative, or take a shortcut via
16/(x+5)^2 > 1/(2x+3)^2
16(2x+3)^2 > (x+5)^2
64x^2 + 192x + 144 > x^2 + 10x + 25
63x^2 + 182x + 119 > 0
9x^2 + 26x + 13 > 0
(9x+17)(x+1) > 0
x is in (-∞,-17/9) U (-1.∞)
But this does not take into account the fact that x ≠ -5 , -3/2
So the real solution is
(-∞,-5) U (-5,-17/9) U (-1.∞)