solve
x+21/x+3<2
x+21/x+3 - 2(x+3)/x+3 <0
x+21-2(x-3)/x+3<0
x+21-2x-6/x+3<0
answer
15-x/x-3<0(is this right?)
You are close. They want you to find the values of x so that the equation the inequality is true. So just do that and you have your answer. I'll be happy to check you answer again when you are done.
Just a note. When you are writing fractions horizontally, you have to put parenthesis around the numerator and the denominator. Otherwise like for your x+21/x+3
It's the same as x + (21/x) + 3. The proper way to do it is (x+21)/(x+3) but anyways its very minor and you probably don't right fractions horizontally in real life, normally.
Is this right?
(15-x)/(x-3)<0
15-x<0
-x<-15
answer is x>15
x+3<0
answer is x<-3
To solve the inequality x + 21/(x + 3) < 2, let's break it down step by step:
1. We first observe that the expression has a fraction in it. To simplify it, we can find a common denominator for the terms. In this case, the common denominator is (x + 3).
So, rewriting the expression with the common denominator:
(x(x + 3)/(x + 3) + 21/(x + 3)) < 2
Simplifying further:
(x(x + 3) + 21)/(x + 3) < 2
2. Next, we multiply each term in the inequality by (x + 3) to eliminate the fraction:
[(x^2 + 3x) + 21] < 2(x + 3)
Expanding and simplifying:
x^2 + 3x + 21 < 2x + 6
3. Rearranging the terms to form a quadratic inequality:
x^2 + 3x - 2x + 21 - 6 < 0
Simplifying:
x^2 + x + 15 < 0
4. At this point, we have the quadratic inequality x^2 + x + 15 < 0.
To determine the solution, we can either use factoring, completing the square, or the quadratic formula. However, in this case, the quadratic does not factor nicely and completing the square might be quite involved.
Instead, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
For our equation, the coefficients are a = 1, b = 1, and c = 15.
x = (-(1) ± √((1)^2 - 4(1)(15)))/(2(1))
x = (-1 ± √(1 - 60))/2
x = (-1 ± √(-59))/2
Since the discriminant (√(-59)) is negative, the quadratic has no real solutions.
Therefore, x^2 + x + 15 < 0 has no solutions.
Hence, the answer to the inequality x + 21/(x + 3) < 2 is x ∈ R.