How would I solve: x-1/x+2 is greater than or equal to x/x-2
Subtract the right side from both sides, leaving zero on the right side.
[(x-1)/(x+2)] - [x/(x-2)] >= 0
Get common denominator.
Multiply first term by (x-2)/(x-2).
Multiply second term by (x+2)/(x+2).
[(x-2)(x-1)]/[(x-2)(x+2)] - [(x+2)x]/[(x+2)(x-2)] >= 0
Multiply.
(x^2 - x - 2x + 2)/(x^2 - 4) - (x^2 + 2x)/(x^2 - 4) >= 0
Combine the terms into one rational expression:
(x^2 - 3x + 2 - x^2 - 2x)/(x^2 - 4) >= 0
Simplify:
(-5x + 2)/(x^2 - 4) >= 0
Find the value of x that will satisfy the equality part of the inequality.
Determine what value of x will make this true:
(-5x + 2)/(x^2 - 4) = 0
To do this, determine what value of x will make numerator equal to zero:
-5x + 2 = 0
-5x + 2 - 2 = 0 - 2
-5x = -2
x = 2/5
Thus, when x=2/5, the rational expression on the right side will be equal to zero.
x = 2/5 is a solution
Determine which value(s) of x will make the rational expression undefined.
To do this, determine what value of x will make denominator equal to zero:
x^2 - 4 = 0
x^2 = 4
x = 2, x = -2
Thus, when x = 2 or x = -2, the rational expression on the right side will be undefined.
x = 2 and x = -2 are not solutions
So far, we know that
x = -2 is not a solution
x = 2/5 is a solution
x = 2 is not a solution.
We need to choose four values for x to represent all the possible values of x to determine all the valid solutions.
Choose an x value less than -2
Choose an x value between -2 and 2/5
Choose an x value between 2/5 and 2
Choose an x value greater than 2
Choose an x value less than -2: x = -3
Substitute this into the inequality:
(-5x + 2)/(x^2 - 4) >= 0
(-5(-3) + 2)/((-3)^2 - 4) >= 0
(15 + 2)/(9 - 4) >= 0
17/5 >= 0 is true
So x values less than -2 are solutions.
Choose an x value between -2 and 2/5: x = -1
Substitute this into the inequality:
(-5x + 2)/(x^2 - 4) >= 0
(-5(-1) + 2)/((-1)^2 - 4) >= 0
(5 + 2)/(1 -4) >= 0
7/-3 >= 0 or -7/3 >= 0 is not true
So x values beteen -2 and 2/5 are not solutions.
Choose an x value between 2/5 and 2: x = 1
Substitute this into the inequality:
(-5x + 2)/(x^2 - 4) >= 0
(-5(1) + 2)/((1)^2 - 4) >= 0
(-5 + 2)/(1 - 4) >= 0
-3/-3 >= 0 or 1 >= 0 is true
So x values between 2/5 and 2 are solutions.
Choose an x value greater than 2: x = 3
Substitute this into the inequality:
(-5x + 2)/(x^2 - 4) >= 0
(-5(3) + 2)/((3)^2 - 4) >= 0
(-15 + 2)/(9 - 4) >= 0
-13/5 >= 0 is not true.
So x values greater than 2 are not solutions.
So we now know that:
x values less than -2 are solutions
x = -2 is not a solution
x values beteen -2 and 2/5 are not solutions
x = 2/5 is a solution
x values between 2/5 and 2 are solutions
x = 2 is not a solution
x values greater than 2 are not solutions
Therefore the solutions for this inequality are represented as:
-infinity < x < -2 and 2/5 <= x < 2
(-infinity, -2) and [2/5, 2)
Note: small correction to my previous answer...
Both occurrences of "rational expression on the right side" should be replaced with just "rational expression".
To solve the inequality (x - 1)/(x + 2) ≥ x/(x - 2), we need to find the values of x for which the inequality holds true. Here's how to solve it step by step:
Step 1: Identify the restrictions
Since we have fractions in the inequality, we need to consider the denominators. In this case, x cannot be equal to -2 or 2 because it would make the denominators equal to zero, resulting in undefined values. So, we must exclude x = -2 and x = 2 from our solution set.
Step 2: Simplify the inequality
To simplify the inequality, we first need to get rid of the denominators. We can do this by multiplying both sides by common denominators. In this case, the common denominator is (x + 2)(x - 2).
Recall that when multiplying an inequality by a negative number, the direction of the inequality flips. Therefore, we need to consider both possibilities when multiplying by (x + 2)(x - 2).
Multiplying the left side:
(x + 2)(x - 2) * [(x - 1)/(x + 2)] ≥ (x + 2)(x - 2) * (x/(x - 2))
Cancel out the (x + 2) terms:
(x - 2)(x - 1) ≥ x(x + 2)
Step 3: Expand and simplify the inequality
Now let's expand both sides of the inequality:
(x - 2)(x - 1) ≥ x(x + 2)
(x^2 - 3x + 2) ≥ (x^2 + 2x)
Simplifying further:
x^2 - 3x + 2 ≥ x^2 + 2x
Subtracting x^2 from both sides:
-3x + 2 ≥ 2x
Step 4: Rearrange the inequality
To solve for x, we need the variable term on one side and constant term on the other. Let's rearrange the inequality:
-3x - 2x ≥ -2
Combining like terms:
-5x ≥ -2
Step 5: Solve for x
To isolate x, we divide both sides of the inequality by -5. However, since we are dividing by a negative number, we need to flip the direction of the inequality:
x ≤ 2/5
Step 6: Finalize the solution
The last step is to take into account the restriction we identified in Step 1. Since x cannot be equal to -2 or 2, the solution set is:
x ≤ 2/5, where x ≠ -2, 2
This means that any value of x less than or equal to 2/5 (excluding -2 and 2) satisfies the original inequality.