Solve the inequality algebraically:
5/x-3 + 3/2-x > 1
(greater then or equal to one)
I know how to do it in theory but I keep getting the answer of just y=2 and I know that can't be right.
Help is really appreciated! Thank you
I am sure you mean:
5/(x-3) + 3/(2-x) ≥ 1
5/(x-3) + 3/(2-x) - 1 ≥ 0
(5(2-x) + 3(x-3) - (x-3)(2-x) ≥ 0 , skipping common denominator for easier typing
10 - 2x + 3x - 9 + (x-3)(x-2) ≥ 0
1 + x + x^2 - 5x + 6 ≥ 0
(x^2 -7x + 7)/((x-3)(2-x)) ≥ 0 , brought back the LCD
The quadratic does not factor, but I found x = 1.2 and x = 5.8 to be approximate solutions.
then checking for the different intervals, we have
1.2 ≤ x < 2 OR 3 < x ≤ 5.8
To solve the inequality algebraically, follow these steps:
Step 1: Find the common denominator for the fractions involved. In this case, the common denominator is (x - 3)(2 - x).
Step 2: Rewrite the equation using the common denominator:
(5(x - 2) + 3(x - 3))/(x - 3)(2 - x) > 1
Simplifying, we get:
(5x - 10 + 3x - 9)/(x - 3)(2 - x) > 1
(8x - 19)/(x - 3)(2 - x) > 1
Step 3: Multiply both sides of the inequality by the denominator to eliminate it:
((x - 3)(2 - x)) * (8x - 19)/(x - 3)(2 - x) > 1 * (x - 3)(2 - x)
8x - 19 > (x - 3)(2 - x)
Step 4: Expand the right side of the equation:
8x - 19 > 2x - 5 - x^2 + 3x
Simplifying further:
8x - 19 > 2x - x^2 + 3x - 5
Combine like terms:
8x - 19 > 5x - x^2 - 5
Step 5: Rearrange the equation to bring all terms to one side:
x^2 + 3x - 5x - 8x + 5 - 19 > 0
x^2 - 10x - 14 > 0
Step 6: Solve the quadratic equation:
To find the values of x that make the inequality true, we can factor the quadratic equation or use the quadratic formula:
Factoring as:
(x - 7)(x + 2) > 0
The zeros of the equation are x = 7 and x = -2.
We can now create a number line and test intervals to see when the expression (x - 7)(x + 2) is greater than zero. On the intervals where it is positive, the inequality is true.
Using test numbers, we can plug in values in each interval:
For x < -2, for example, we can take x = -3 and get (-3 - 7)(-3 + 2) = (-10)(-1) = 10 > 0. So it is true for x < -2.
For -2 < x < 7, we can take x = 0 and get (0 - 7)(0 + 2) = (-7)(2) = -14 < 0. So it is false for -2 < x < 7.
For x > 7, we can take x = 8 and get (8 - 7)(8 + 2) = (1)(10) = 10 > 0. So it is true for x > 7.
Therefore, the solution to the inequality is x < -2 or x > 7.
Please note that if you received y = 2 as the answer, there might be an error in your calculations or interpretation of the inequality.
To solve the inequality algebraically, we need to find the values of x that satisfy the given inequality. Let's start by simplifying the expression:
5/(x-3) + 3/(2-x) > 1
To remove the fractions, we can find a common denominator. The common denominator in this case is (x-3)(2-x). Let's multiply every term by this common denominator:
5(x-3)(2-x)/(x-3) + 3(x-3)(2-x)/(2-x) > (x-3)(2-x)
Now, let's simplify each term:
5(2-x) + 3(x-3) > (x-3)(2-x)
10 - 5x + 3x - 9 > 2x - 6 - x^2 + 3x
Combining like terms, we get:
4 - 2x > -x^2 + 5x - 6
Rearranging the terms to bring all terms to one side of the inequality, we have:
0 > -x^2 + 7x - 10
Now, let's rearrange the equation to set it equal to zero:
x^2 - 7x + 10 < 0
To solve this quadratic inequality, we can factor it:
(x - 5)(x - 2) < 0
Now, let's analyze the sign of the inequality in different intervals:
When x < 2:
Both factors are negative, so the product is positive. This interval is not part of the solution.
When 2 < x < 5:
The first factor is positive, and the second factor is negative. The product is negative. This interval satisfies the inequality.
When x > 5:
Both factors are positive, so the product is positive. This interval is not part of the solution.
Therefore, the solution to the inequality is:
2 < x < 5
So, the answer is not just x = 2; it is the range of values between 2 and 5, exclusive.
Remember to always check the solutions by substituting them back into the original inequality to ensure they satisfy the given constraint.