the complete solution set of x which satisfy the inequality
[4/(x-2) ]>=[3/(2x-1)]
By cross-multiplying the inequality.
8x-4>=3x+2
5x>=-2
Yes, you are correct. That is another way to solve the inequality.
Well, well, well, looks like we've got ourselves an inequality party here! Let's dive in and solve this bad boy!
First, we want to get rid of those pesky fractions, so we'll multiply both sides of the inequality by the least common denominator, which is (x-2)(2x-1). That way, the fractions will cancel out, and we can focus on the good stuff.
4(x-2)(2x-1) ≥ 3(x-2)
Now let's distribute and simplify a bit:
8x^2 - 4x - 4x + 2 ≥ 3x - 6
Combine like terms:
8x^2 - 8x + 2 ≥ 3x - 6
Bring everything to one side of the inequality:
8x^2 - 11x + 8 ≥ 0
Now, here's where it gets interesting. To find the solutions, we need to factor this quadratic expression. But hang on a second, it doesn't seem like it can be factored nicely. Uh-oh!
So, instead, we'll draw a little graph and see where this inequality is greater than or equal to zero.
(Inserts a colorful graph)
Hmm, it seems like the solution set includes all those values of x where the graph is above or equal to the x-axis. But we don't want to leave any stone unturned, so let's find those critical points where the graph intersects the x-axis.
To find the critical points, we set the expression equal to zero:
8x^2 - 11x + 8 = 0
But just like before, this quadratic equation doesn't play nicely with others, so we'll have to use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Plug in the values of a, b, and c, and calculate those critical points. I'll wait.
(Drumming fingers)
Alright, so we've got some critical points. Now, remember, we want those values of x where the graph is greater than or equal to zero. So, let's put it all together:
The solution set is x ≤ (the first critical point calculated) OR x ≥ (the second critical point calculated) OR (the first critical point calculated) ≤ x ≤ (the second critical point calculated).
Phew, we made it! So, there you have it, folks, the complete solution set of x that satisfies the inequality. Enjoy the graph party!
To find the solution set for the inequality [4/(x-2)] >= [3/(2x-1)], we need to solve it step by step. Here's how:
Step 1: Remove the denominators by multiplying both sides of the inequality by the least common denominator (LCD) of (x-2) and (2x-1). The LCD is (x-2)(2x-1), so we multiply both sides by (x-2)(2x-1):
[4/(x-2)] * (x-2)(2x-1) >= [3/(2x-1)] * (x-2)(2x-1)
Step 2: Simplify the equation by canceling out common factors:
4(2x-1) >= 3(x-2)
Step 3: Expand and simplify the equation further:
8x - 4 >= 3x - 6
Step 4: Move all variables to one side of the inequality by subtracting 8x and adding 6 on both sides:
8x - 3x >= -4 + 6
5x >= 2
Step 5: Divide both sides of the inequality by 5 to solve for x:
x >= 2/5
Therefore, the complete solution set for the inequality [4/(x-2)] >= [3/(2x-1)] is x >= 2/5.
watch out for zero in denominators
[4/(x-2) ]>=[3/(2x-1)]
must not include x = 2 or x = 1/2
[4(2x-1)]>=[3(x-2)]
8 x - 4 >/= 3 x - 6
5 x >/= -2
x >/ = -2/5 but exclude x = 2 and x = 1/2
Is [] supposed to be absolute value? If so then
|4/(x-2)| >= |3/(2x-1)|
has solution set
x <= -2/5 or x > 2