3.4 Solve Rational Equations and Inequalities
a)4/3x-5=4
b)x-5/x^2-3x-4=3x+2/x^2-1
4=4(3x-5), x!=5/3
x-5/(x-4)(x+1) =3x+2/(x-1)(x+1), x!=-1, x!=1, x!=4
4=12x-20
(x-5)(x-1)=(3x+2)(x-4)
12x=24
x =2
x=4+(or)-√(-4)^2-4(2)(-13)/2(2)
=4+(or)-√120/4
=4+(or)-2√30/4
=2+(or)-√30/2
x^2-6x+5=3x^2-10x-8
2x^2-4x-13=0
x/x-2=2x^2-3x+5/x^2+6
x(x^2+6)=(x-2)(2x^2-3x+5), x!=2
x^3+6x=2x^3-7x^2+11x-10
x^3-7x^2+5x-10=0
x=6.47
Method 1: Consider the key features of the graph of a Related Rational function
2/x-5<10
2/x-5-10<0
2-10x+50/x-5<0
-10x-52/x-5<0
The vertical asymptote: x=5
The horizontal asymptote: y= -10
The x-intercept is 5.2.
The y-intercept is –10.4.
So, -10x-52/x-5<0, or 2/x-5<10
For x<5 or x>5.2.
In interval notation, xe(-∞,5)U(5.2, +∞)
Method 2: Solve Algebraically
2/x-4<10
Because x-5!=0, either x>5 or x<5. (x>|5|)
Case 1: when x>5
2/x-5<10 ---> 2<10(x-5)
2<10x-50
52<10x
x>5.2
X>5.2 is within x>5, so the solution is x>5.2.
Case 2: when x<5
2.x-5<10 ---> 2>10(x-5)
2>10x-50
52>10x
x<5.2
x<5 is within x<5.2, so the solution is x<5.
Combining the two cases, the solution to the inequality is x<5 or x>5.2.
Method 1: Using an Intercal Table
x^2-x-2/x^2+x-12>=0
(x-2)(x+1)/(x-3)(x+4)>=0
From the numerator, the zeros occur at x=2 and x=-1, so solutions occur at these values of x.
From the denominator, the restriction occur at x= 3 and x=-4, A number line can be used to consider
intervals.
Interval
Choice for x in interval
(-∞,-4)
(-4,-1)
-1
(-1,2)
2
(2,3)
(3,+∞)
x=-5
x=-2
x=1
x=0
x=2
x=2.5
x=4
Sings of factors of
(x-2)(x+1)/(x-3)(x+4)
(-)(-)/(-)(-)
(-)(-)/(-)(+)
(-)(0)/(-)(+)
(-)(+)/(-)(+)
(0)(+)/(-)(+)
(+)(+)/(-)(+)
(+)(+)/(+)(+)
Sign of
(x-2)(x+1)/(x-3)(x+4)
+
0
+
0
+
For the inequality x^2-x-2/x^2+x-12>=0, the solution is x< -4 or –1<=x<=2 or x>3.
In interval notation, the solution set is xe(-∞,-4)U[-1,2]U(3,+∞)