When solving rational inequalities, avoid cross-multiplying as it can cause you to miss critical points (vertical asymptotes) where the denominator equals zero. Instead, move all terms to one side, combine into a single fraction, find critical points by setting both numerator and denominator to zero, and test intervals to determine where the expression is positive. For the inequality (x²-1)/(x²-1) > 1, the solution is x ∈ (-√3, -1) ∪ (1, √3).
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Do NOT Cross-Multiply This! 🛑 (Rational Inequalities Trick) #mathsshorts #shorts #maths #mathematics追加:
So, this question asks us to solve the inequality given to us.
First, we can subtract one from both sides before simplifying the left side of the inequality into a single fraction.
Rewriting minus one as x squared minus one divided by itself.
Then, with a common denominator of x squared minus one, simplify the numerator to get three minus x squared.
We need values of x that are equal to zero.
So, we need to set the expressions three minus x squared and x squared minus one equal to zero before solving to find the range of values for x, which are plus or minus root three and plus or minus one, respectively.
There are five sections of the graph where the sign of the y values needs to be found.
The inequality asks us to solve for positive values and those values are when x is between minus root three and minus one and one and root three.
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