Solve the following inequality. 4x-4/x+3≤3
I will assume you mean
(4x-4)/(x+3) ≤ 3
then 4x-4 ≤ 3x + 9
x ≤ 13
as confirmed by Wolfram
http://www.wolframalpha.com/input/?i=plot++%284x-4%29%2F%28x%2B3%29+≤+3
To solve the inequality 4x - 4/(x + 3) ≤ 3, we will follow these steps:
Step 1: Find the domain.
The inequality involves a fraction, so we need to make sure the denominator (x + 3) is not equal to zero since division by zero is undefined. Therefore, the domain is all real numbers except x = -3.
Step 2: Multiply through by the denominator (x + 3).
When dealing with fractions in inequalities, it is often easier to eliminate the denominator by multiplying both sides of the inequality by the common denominator. In this case, the common denominator is (x + 3). Multiplying through gives us:
(x + 3) * (4x - 4) / (x + 3) ≤ 3 * (x + 3)
This simplifies to:
4x - 4 ≤ 3x + 9
Step 3: Solve the resulting equation.
To solve this equation, we will gather the x terms on one side and the constant terms on the other side.
4x - 3x ≤ 9 + 4
Simplifying further:
x ≤ 13
Step 4: Check the inequality domain.
We need to check the domain again since we multiplied both sides by (x + 3). The inequality is valid for all real numbers except x = -3. Therefore, the solution is:
x ≤ 13, x ≠ -3
So the inequality 4x - 4/(x + 3) ≤ 3 is satisfied for all values of x less than or equal to 13, excluding x = -3.