I need to solve this inequality:
3(x-1)^2 > 0
The answer is x �‚ 1, but I don't understand why the answer is not allowed to be equal to 1?
Thanks :)
Sorry, the symbol did not come out properly when I posted the question. The answer to the question is x followed by an equals sign with a line going through it then 1.
Look at the expression
(x-1)^2
When you square anything, the result is always positive unless you squared zero, which would stay at zero.
3(x-1)^2 > 0
divide by +3
(x-1)^2 > 0
now for what value is x-1 equal to zero?
x-1 = 0
x = 1
now take any value of x you want, positive or negative, take 1 away and square you will always get a positive answer.
EXCEPT, when x = 1, we get 0^2 > 0 which is false, so ....
the solution would be
x ∈ R, x ≠ 1
if your question had been
3(x-1)^2 ≥ 0, then it would include x=1
and x ∈ R
Thank you for answering, it really helped :)
Just wondering when I am supposed to know to put x �¸ R?
Thanks
To solve the inequality 3(x-1)^2 > 0, we can follow these steps:
Step 1: Begin by recognizing that the inequality involves a quadratic expression, (x-1)^2. In general, a quadratic expression is always non-negative, meaning it is equal to or greater than zero. Therefore, (x-1)^2 could be zero or positive.
Step 2: Next, we introduce the coefficient 3. Since 3 multiplied by any non-negative value will yield a non-negative result, multiplying (x-1)^2 by 3 will either yield zero or a positive value.
Step 3: Now, let's solve the equation (x-1)^2 = 0 and determine the solutions. Taking the square root of both sides, we have:
(x-1) = 0
x = 1
Thus, the equation (x-1)^2 = 0 has the solution x = 1.
Step 4: Finally, we need to determine the values of x for which 3(x-1)^2 > 0. Since we already know that (x-1)^2 will either be zero or positive, multiplying it by 3 guarantees that the entire expression, 3(x-1)^2, will be positive for all values of x except x = 1.
Therefore, the solution to the inequality 3(x-1)^2 > 0 is x ≠ 1, which means x can be any real number except 1. The inequality excludes x = 1 because 3(x-1)^2 is greater than zero for all other values of x, but equal to zero at x = 1.