what can x equal if
4 < 2x+1/x-3 < 7
multiply by x-3
4(x-3) < 2x + 1 < 7(x-3) , if x-3 > 0 or x > 3
4x-12 < 2x+1 or 2x+1 < 7x - 21
2x < 13 ----- or -5x < -22
x < 6.5 ------ or x > 22/5
So we have 3 critical values, x = 6.5 , x = 4.4 and x = 3
dividing our number line into 4 segments.
I then take any value in each of the segments and test it in the original
left segment , x < 3, e.g let x = 0
4 < 1/-3 < 7 false
segment between 3 and 4.4, x = 4
4 < 9/1 < 7 false
segment between 4.4 and 6.5, x = 5
4 < 11/2<7
4<5.5<7 true
segment > 6.5, x = 10
4 < 21/8<7
4 < 2.625<7 false
so 4.4 < x < 6.5
btw, if you have a graphing calculator
graph
y = (2x+1)(x-3) , y = 4 and y = 7 and you will see that they intersect at (4.4 , 7) and (6.5 , 4)
so the graph is between 4 and 7 for my values
To find the possible values of x that satisfy the given inequality, let's break it down into two separate inequalities and solve them individually.
First, let's consider the inequality 4 < 2x + 1/(x - 3). We need to get rid of the fraction, so we can multiply the entire inequality by (x - 3) to eliminate the denominator:
4(x - 3) < 2x(x - 3) + 1
Simplifying this expression, we get:
4x - 12 < 2x^2 - 6x + 1
Rearranging the terms, we have:
2x^2 - 6x + 1 - 4x + 12 > 0
2x^2 - 10x + 13 > 0
Now, let's solve this quadratic inequality. We could use the quadratic formula or factoring, but let's try factoring first. Unfortunately, the quadratic cannot be factored easily.
So let's consider the discriminant (b^2 - 4ac) to determine the nature of the solutions. In this case, a = 2, b = -10, and c = 13.
The discriminant (b^2 - 4ac) = (-10)^2 - 4 * 2 * 13 = 100 - 104 = -4
Since the discriminant is negative, the quadratic does not have real solutions. Therefore, there are no solutions to the inequality 4 < 2x + 1/(x - 3).
Now, let's move on to the second inequality: 2x + 1/(x - 3) < 7.
We'll apply the same approach as before. Multiply both sides of the inequality by (x - 3) to eliminate the fraction:
2x(x - 3) + 1 < 7(x - 3)
2x^2 - 6x + 1 < 7x - 21
Rearranging the terms, we get:
2x^2 - 13x + 22 < 0
Again, let's check the discriminant:
The discriminant (b^2 - 4ac) = (-13)^2 - 4 * 2 * 22 = 169 - 176 = -7
Since the discriminant is negative, the quadratic does not have real solutions. Therefore, there are no solutions to the inequality 2x + 1/(x - 3) < 7.
In conclusion, there are no values of x that satisfy the given inequality 4 < 2x + 1/(x - 3) < 7.