how do you solve this inequality
3x-2 < x+4/x-2
please help i don't get it
You must mean:
3x-2 < (x+4)/(x-2) , so clearly x ≠ 2
(x-2)(3x-2) < x+4
3x^2 - 8x + 4 - x - 4 < 0
3x^2 - 9x < 0
3x(x-3) < 0
So the "critical values" are x=0 , x = 2, and x=3
test for a value of x<0
let x=-5 in original
-17 < -1/-7
-17 < 1/7 ? YES
test for a value between 0 and 2
let x=1
1 < 5/-1
1 < -5 ? NO
test for a value between 2 and 3
let x = 2.5
5.5 < 6.5/.5
5.5 < 13 ? , YES
test for a value x>3
let x=5
13 < 9/3
13 < 3 ? , NO
so we have x ≤ 0 OR 2 < x ≤ 3 , notice that x=2 in not included
To see that this is correct
go to my favourite graphing program
http://rechneronline.de/function-graphs/
and enter
3x-2 - (x+4)/(x-2)
in the window for "first graph"
change the "Range y-axis from" entries to -200 to 200
you will see the curve below the x-axis from -infinitity to 0
above the x-axis from 0 to the asymptote of 2
below the x-axis from 2 to 3, and
above the x-axis for x>3
thanks so much :)
what if all the values you put in do not work. how would you write the intervals.?
If the inequality can not be satisfied for any real value of x, then you just have to say that.
To solve the inequality 3x - 2 < (x + 4) / (x - 2), we can follow these steps:
Step 1: Multiply both sides of the inequality by (x - 2) to clear the fraction. Remember, when multiplying both sides of an inequality by a negative number, the direction of the inequality is reversed.
(x - 2)(3x - 2) < (x + 4)
Step 2: Expand the left side of the inequality by multiplying (x - 2) with each term within the parentheses.
3x^2 - 2x - 6x + 4 < x + 4
Simplifying further:
3x^2 - 8x + 4 < x + 4
Step 3: Rearrange the terms by moving x and the constants to the left side, so the inequality becomes:
3x^2 - 8x - x - 4 + 4 < 0
3x^2 - 9x < 0
Step 4: Factor out an x from the left side:
x(3x - 9) < 0
Step 5: Determine the critical points by setting each factor equal to zero and solving for x:
x = 0
3x - 9 = 0
3x = 9
x = 3
Step 6: Create a sign chart or number line and choose test values within the intervals created by the critical points. Plug in these values into (3x - 9) to determine the signs.
Testing x = -1: (-1)(-3 - 9) < 0 => 12 < 0 (FALSE)
Testing x = 1: (1)(3 - 9) < 0 => -6 < 0 (TRUE)
Testing x = 4: (4)(12 - 9) < 0 => 12 < 0 (FALSE)
Step 7: Analyze the results from the sign chart. The solution to the inequality is the intervals where the expression x(3x - 9) < 0 evaluates to true (negative).
Based on the sign chart provided, we can deduce that x is between 0 and 3 because that interval satisfies the inequality.
In interval notation, the solution is: 0 < x < 3