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Question 345416: I have to solve the following nonlinear inequality and provide the solution in interval notation.
1) -x^2<-7x
2)x<5x+6/x
Answer by haileytucki(390)   (Show Source):
You can put this solution on YOUR website! -x^(2)<-7x (any ~ seen mean that it is a square root sign)
Since -7x contains the variable to solve for, move it to the left-hand side of the inequality by adding 7x to both sides.
-x^(2)+7x<0
Multiply each term in the equation by -1.
x^(2)-7x>0
Factor out the GCF of x from each term in the polynomial.
x(x)+x(-7)>0
Factor out the GCF of x from x^(2)-7x.
x(x-7)>0
Set the single term factor on the left-hand side of the inequality equal to 0 to find the critical points.
x=0
Set each of the factors of the left-hand side of the inequality equal to 0 to find the critical points.
x-7=0
Since -7 does not contain the variable to solve for, move it to the right-hand side of the equation by adding 7 to both sides.
x=7
To find the solution set that makes the expression greater than 0, break the set into real number intervals based on the values found earlier.
x<0_0
Determine if the given interval makes each factor positive or negative. If the number of negative factors is odd, then the entire expression over this interval is negative. If the number of negative factors is even, then the entire expression over this interval is positive.
x<0 makes the expression positive_0
Since this is a 'greater than 0' inequality, all intervals that make the expression positive are part of the solution.
x<0 or x>7
x<5x+(6)/(x)
Since x is on the right-hand side of the equation, switch the sides so it is on the left-hand side of the equation.
5x+(6)/(x)>x
To set the left-hand side of the inequality equal to 0, move all the expressions to the left-hand side.
5x+(6)/(x)-x>0
Since 5x and -x are like terms, add -x to 5x to get 4x.
4x+(6)/(x)>0
To add fractions, the denominators must be equal. The denominators can be made equal by finding the least common denominator (LCD). In this case, the LCD is x. Next, multiply each fraction by a factor of 1 that will create the LCD in each of the fractions.
4x*(x)/(x)+(6)/(x)>0
Complete the multiplication to produce a denominator of x in each expression.
(4x^(2))/(x)+(6)/(x)>0
Combine the numerators of all expressions that have common denominators.
(4x^(2)+6)/(x)>0
Factor out the GCF of 2 from each term in the polynomial.
(2(2x^(2))+2(3))/(x)>0
Factor out the GCF of 2 from 4x^(2)+6.
(2(2x^(2)+3))/(x)>0
Find all the values where the expression switches from negative to positive by setting each factor equal to 0 and solving.
x=0_(2x^(2)+3)=0
Solve for each factor to find the values where the absolute value expression goes from negative to positive.
x=0_x=(i~(6))/(2),-(i~(6))/(2)
Find the minimum solution set of the inequality.
x>0
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