5. Rational Equations
In short terms, A rational equation is an equation
with one or more rational expressions. Containing
at least one fraction whose numerator and
denominator are polynomials.
6. What are some examples of Rational
Equations?
π₯
5
β
1
4
=
π₯
2
3 π¦ + 3
π¦ + 1
+ 3 =
3π¦ β’ 1
4 + 1
π₯
2
+
3π₯
5
=
π₯ β 1
4
π₯ β 2
2π₯
β
π₯
π₯ + 1
=
4 1 β π₯
6π₯
β’ Take note that a rational equation is an equation that have two
or more Rational Expressions or a polynomial on its numerator
and denominator.
8. How to solve Rational Equations?
β’ To solve rational equations you will need to follow
these (4) four steps:
Find the LCD
Multiply everything by the LCD
Simplify
Checking of Answers
9. In this problem, we will solve this
3-R. expression, rational equation.
4π₯
8π₯ + 1
+
14
3π₯
=
46
9π₯
10. STEP 1:
Find the LCD
4π₯
8π₯ + 1
+
14
3π₯
=
46
9π₯
First, we would need to find the LCD of our
equation.
11. STEP 1:
Find the LCD
4π₯
8π₯ + 1
+
14
3π₯
=
46
9π₯
To find the LCD we must search for terms with
the least similar multiplies found in the
denominator.
12. STEP 1:
Find the LCD
4π₯
8π₯ + 1
+
14
3π₯
=
46
9π₯
In our case, the common terms found in this
equation will be 9x.
13. STEP 1:
Find the LCD
4π₯
8π₯ + 1
+
14
3π₯
=
46
9π₯
Why? Because in the multiples of 3x and 9x. If
you will interpret their common factor.
14. STEP 1:
Find the LCD
4π₯
8π₯ + 1
+
14
3π₯
=
46
9π₯
9x, would become their common factors or in
other terms the LCD.
15. STEP 1:
Find the LCD
4π₯
8π₯ + 1
+
14
3π₯
=
46
9π₯
But what would happen with the polynomial 8x +1?
16. STEP 1:
Find the LCD
4π₯
8π₯ + 1
+
14
3π₯
=
46
9π₯
Since it is a binomial, it is not applicable to find the
LCD because it has no similar expressions present.
17. STEP 1:
Find the LCD
9x (8x+1)(
4π₯
8π₯+1
+
14
3π₯
=
46
9π₯
)
So instead, we would copy the binomial and place it
beside the LCD of 3x and 9x.
18. STEP 2:
Multiply everything by the LCD
9x (8x+1)(
4π₯
8π₯+1
+
14
3π₯
=
46
9π₯
)
To multiply we can use the distributive property or
cancellation.
19. STEP 2:
Multiply everything by the LCD
9x (8x+1)(
4π₯
8π₯+1
+
14
3π₯
=
46
9π₯
)
To multiply we can use the distributive property or
cancellation.
20. STEP 2:
Multiply everything by the LCD
9x (8x+1)(
4π₯
8π₯+1
+
14
3π₯
=
46
9π₯
)
But to answer our problem we will be using the
cancellation method because it is easier and faster.
21. STEP 2:
Multiply everything by the LCD
9x (8x+1)(
4π₯
8π₯+1
+
14
3π₯
=
46
9π₯
)
To Cancel, cancel the denominators with similar or
factor terms
22. STEP 2:
Multiply everything by the LCD
4π₯
9x
+
14
3x(8x + 1)
=
46
8x + 1
After cancelling you will multiply the denominator to
the numerator.
23. STEP 2:
Multiply everything by the LCD
(9x)(4x) + 3(8x+1)(14) = 46 (8x +1)
Rewrite the terms and then multiply.
24. STEP 3:
36x2 + 336x + 42 = 368x + 46
After multiplying, you will now simplify the equation
firstly by transposition or the transfer of terms to
another.
Simplify
36x2 + 336x -368x -46 + 42= 0
25. STEP 3:
After transposing the terms we will be going to
combine the like terms.
Simplify
36x2 + 336x -368x -46 + 42= 0
36x2 -32x -4 = 0
26. STEP 3:
Now, we got our answer from combining like terms.
Usually, the answer will the equal to x but if the
answer shows to be a quadratic equation.
Simplify
36x2 -32x -4 = 0
27. STEP 3:
Simplify
To solve a quadratic equation, we are going to be
using the Quadratic Formula.
36x2 -32x -4 = 0
π =
βπ Β± ππ β πππ
ππ
28. STEP 3:
Simplify
Substitute the terms of ax2 + bx + c = 0, to the
formula and then simplify.
36x2 -32x -4 = 0
π =
β(βππ) Β± (βππ)πβπ(ππ)(βπ)
π(ππ)
31. STEP 4:
Checking of Answers
After doing the Quadratic Formula and got our
answer, in this part we will going to be checking if
our solution is TRUE or FALSE (extraneous root)
ππ = π ππ = β
π
π
32. STEP 4:
Checking of Answers ππ = π
4π₯
8π₯ + 1
+
14
3π₯
=
46
9π₯
To check our answers, we will be going to
substitute the value of x and then simplify the
equation.
4(1)
8(1) + 1
+
14
3(1)
=
46
9(1)
34. STEP 4:
Checking of Answers
4π₯
8π₯ + 1
+
14
3π₯
=
46
9π₯
To check our answers, we will be going to substitute the
value of x and then simplify the equation.
4(β
1
9
)
8(β
1
9
) + 1
+
14
3(β
1
9
)
=
46
9(β
1
9
)
ππ = β
π
π
36. CONCLUSION
οΆ After we have solved the equation, we have found
out that the rational equation given is TRUE. This
means that it has a real solution.
οΆ But you must take note! If a given solution is not real
or β and has a 0 on its denominator. Then it is called
an extraneous root
37. Awesome Words!
βMath is easy when it is Fun!
Donβt stress yourself and relax,
every solution has an answerβ
βL.G. Writings