Solve 1-x/x+5>2-x/x+6 without the use of a graphing calculator show all your work.
the way you typed it, the statement would simply become:
1-x/x+5>2-x/x+6
1 - 1 + 5 > 2 - 1 + 6
5 >7 , which is of course false
so you must have meant:
(1-x)/(x+5) > (2-x)/(x+6) , x ≠ -5,-6
(1-x)(x+6) > (x+5)(2-x)
-x^2 - 5x + 6 > -x^2 - 3x + 10
-2x > 4
x < -2 , x ≠ -5,-6
or
(1-x)/(x+5) > (2-x)/(x+6), x ≠ -5,-6
(1-x)/(x+5) - (2-x)/(x+6) > 0
[(1-x)(x+6) - (2-x)(x+5)]/[(x+5)(x+6)] > 0
-x^2 - 5x + 6 - (-x^2 - 3x + 10) > 0
-2x - 4 > 0
2x + 4 < 0
x < -2 , x ≠ -5,-6
when we get to
-x^2 - 5x + 6 - (-x^2 - 3x + 10) > 0
this is true only if (x+5)(x+6) > 0
That is, x < -6 or x > -5
so, combining that with x < -2, we have a solution set of
(-∞,-6)U(-5,-2)
To solve the inequality 1 - (x / (x+5)) > 2 - (x / (x+6)), we will start by clearing the fractions. Here's a step-by-step solution:
Step 1: Clearing the denominators
Multiply both sides of the inequality by (x+5)(x+6) to eliminate the fractions. Remember that when you multiply by a negative number, you must reverse the inequality sign.
(x+5)(x+6) * (1 - (x / (x+5))) > (x+5)(x+6) * (2 - (x / (x+6)))
Step 2: Expanding the expressions
On the left-hand side, the (x+5) in the numerator cancels with one of the (x+5) in the denominator, leaving just (x+6). On the right-hand side, the (x+6) in the numerator cancels with one of the (x+6) in the denominator, leaving (x+5).
(x+6) - (x+6)(x / (x+5)) > (x+5) - (x+5)(x / (x+6))
Step 3: Simplifying the expressions
Distribute the multiplication on both sides of the inequality:
(x+6) - (x^2 / (x+5)) - (6x / (x+5)) > (x+5) - (x^2 / (x+6)) - (5x / (x+6))
Step 4: Combining like terms
Combine similar terms on both sides of the inequality:
x + 6 - (x^2 / (x+5)) - (6x / (x+5)) > x + 5 - (x^2 / (x+6)) - (5x / (x+6))
Step 5: Transforming the equation
Move all the terms to one side of the inequality:
x + 6 - x - 5 + (x^2 / (x+5)) + (6x / (x+5)) - (x^2 / (x+6)) - (5x / (x+6)) > 0
Step 6: Simplifying further
Combine like terms:
x + 1 + (x^2 / (x+5)) + (x^2 / (x+6)) + (6x / (x+5)) - (5x / (x+6)) > 0
Step 7: Finding the common denominator
To add or subtract fractions, we need a common denominator. In this case, we can use the least common multiple (LCM) as the common denominator for the fractions (x^2 / (x+5)) and (x^2 / (x+6)). The LCM of (x+5) and (x+6) is (x+5)(x+6), so we rewrite the fractions:
x + 1 + ((x^2*(x+6)) / ((x+5)(x+6))) + ((x^2*(x+5)) / ((x+5)(x+6))) + ((6x*(x+6)) / ((x+5)(x+6))) - ((5x*(x+5)) / ((x+5)(x+6))) > 0
Step 8: Simplifying the expression
Combine like terms:
x + 1 + ((x^2*(x+6) + x^2*(x+5) + 6x*(x+6) - 5x*(x+5)) / ((x+5)(x+6))) > 0
Simplify the numerator:
x + 1 + ((x^3 + 6x^2 + 5x^2 + 30x
Continued in the next message...