Download (1) When solving a rational equation, why it is OK to remove the

(1) When solving a rational equation, why it is OK to remove the denominator by multiplying both sides by the LCD
and why can you not do the same operation when simplifying a rational expression? Please show your steps with at
least 75 words.
(2) How do you compute algebraically (not by graphing) the intercepts of any function? For example, what are the xintercept(s) and y-intercept(s) of 2x – 5y = 8 without graphing the equation? Remember that intercepts are points
and points are written as ordered pairs. Please show your steps with at least 75 words.
(3) Solve and show your steps with at least 75 words x/2 +1=x/4
(1) As a rule, we can do any mathematical operation on one side of an equation as long as we do the same on the other
side also. This will keep the equation “intact”. Therefore, we can multiply both sides of a rational equation by the LCD.
For example, consider x/2 + x/5 = 1
Here, the LCD is 10. So we can multiply both sides of the equation by 10 to get
(x/2)(10) + (x/5)(10) = 1(10)
5x + 2x = 10 (Note that the denominators no more exist)
7x = 10
x = 10/7
But we have a rational expression, we don't have an equation and there are no two sides. Therefore, we can't multiply
“both sides” by anything. We can, however, multiply both the numerator and the denominator of the rational
expression by the same non-zero term.
If we had to simplify x/2 + x/7, we could multiply and divide by the LCD, which is 14 to get
(x/2)(14/14) + (x/7)(14/14)
= (7x/14) + (2x/14)
= 9x/14 (The denominator does not vanish)
(2) The intercepts of a function are the points where the graph of the function crosses (or touches) the coordinate axes.
To find the x-intercept(s) of a function algebraically, we plug in y = 0 in the function and solve for x (This is because, at
the point where the graph crosses the x- axis, the y- value is 0)
Similarly, to find the y-intercept(s) of a function algebraically, we plug in x = 0 in the function and solve for y (This is
because, at the point where the graph crosses the y- axis, the x- value is 0)
Consider 2x - 5y = 8
When y = 0, 2x = 8 → x = 4
The x-intercept is (4, 0)
When x = 0, -5y = 8 → y = -8/5
The y-intercept is (0, -8/5)
(3) x/2 + 1 = x/4 is a rational equation
First, collect the x-terms on one side and constant term on the other
x/2 - x/4 = -1
The LCD of 2 and 4 is 4. Multiply both sides by 4
(x/2)(4) - (x/4)(4) = -1(4)
Now, simplify and solve for x
2x - x = -4
x = -4