Solving Rational
Equations
Section 7.5 beginning on page 392
Solving by Cross Multiplying
When you have two equal ratios, their cross products are equal.
Example 1: Solve
3
9
=
𝑥 + 1 4𝑥 + 5
3(4𝑥 + 5) = 9(𝑥 + 1)
12𝑥 + 15 = 9𝑥 + 9
3𝑥 + 15 = 9
3𝑥 = −6
𝑥 = −2
CHECK:
3
9
=
(−2) + 1 4(−2) + 5
3
9
=
−1 −8 + 5
−3 =
9
−3
−3 = −3
Solving by Using the LCD
Example 3a : Solve
4𝑥 5
7 4𝑥
9 4𝑥
∙
+
=− ∙
∙
1 𝑥
4 1
𝑥 1
** Multiply each term by the LCD.
𝐿𝐶𝐷 = 4𝑥
** Check
20𝑥 28𝑥
36𝑥
+
=−
𝑥
4
𝑥
20 + 7𝑥 = −36
7𝑥 = −56
𝑥 = −8
** Simplify
** Solve
5
7 2
9
+ ∙ =−
(−8) 4 2
(−8)
5 14 9
− +
=
8 8
8
−5 + 14 9
=
8
8
9 9
=
8 8
Solving by Using the LCD
Example 3b: Solve
1−
8
3
=
𝑥−5 𝑥
𝑥(𝑥 − 5) ∙ 1 −
𝐿𝐶𝐷 = 𝑥(𝑥 − 5)
8
𝑥(𝑥 − 5) 3 𝑥(𝑥 − 5)
∙
= ∙
𝑥−5
1
𝑥
1
𝑥(𝑥 − 5) −
8𝑥(𝑥 − 5) 3𝑥(𝑥 − 5)
=
𝑥−5
𝑥
𝑥 2 − 5𝑥 − 8𝑥 = 3(𝑥 − 5)
𝑥 2 − 13𝑥 = 3𝑥 − 15
𝑥 2 − 16𝑥 + 15 = 0
𝑥 − 15 𝑥 − 1 = 0
𝑥 = 15, 𝑥 = 1
Solving an Equation With an
Extraneous Solution
Example 4: Solve
6
𝑥−3
6
8𝑥 2
4𝑥
=
−
𝑥 − 3 𝑥2 − 9 𝑥 + 3
8𝑥 2
4𝑥
𝑥 + 3 (𝑥 − 3) =
𝑥 + 3 (𝑥 − 3) −
𝑥 + 3 (𝑥 − 3)
𝑥+3
6 𝑥 + 3 = 8𝑥 2 − 4𝑥(𝑥 − 3)
6𝑥 + 18 = 8𝑥 2 − 4𝑥 2 + 12𝑥
6𝑥 + 18 = 4𝑥 2 + 12𝑥
0 = 4𝑥 2 + 6𝑥 − 18
0 = 2𝑥 2 + 3𝑥 − 9
0 = (2𝑥 − 3)(𝑥 + 3)
3
𝑥= ,
2
𝑥 = −3
𝑥 + 3 (𝑥 − 3)
Solving a Real – Life Problem
50𝑚 + 1000
Example 6 : The function 𝑐 =
represents the average cost c (in
𝑚
dollars) of making m models using a 3-D printer. Find how many models must be
printed for the average cost per model to fall to $90.
𝑚 ∙ 90 =
50𝑚 + 1000
∙𝑚
𝑚
90𝑚 = 50𝑚 + 1000
40𝑚 = 1000
𝑚 = 25
The average cost per model falls to
$90 after 25 have been printed.
Monitoring Progress
Solve each equation and check your solution(s).
1)
3
2
=
5𝑥 𝑥 − 7
3)
1
𝑥
=
2𝑥 + 5 11𝑥 + 8
5)
3𝑥
5
3
−
=
𝑥 + 1 2𝑥 2𝑥
2
𝑥 = − ,𝑥 = 2
3
𝑥 = −3
𝑥 = −1,
𝑥=4
1
3
2)
−4
5
=
𝑥+3 𝑥−3
𝑥=−
4)
15 4 7
+ =
𝑥 5 𝑥
𝑥 = −10
6)
4𝑥 + 1
12
= 2
+3
𝑥+1
𝑥 −1
𝑥 = −2, 𝑥 = 5
Monitoring Progress
Solve each equation and check your solution(s).
7)
9
6𝑥
9𝑥 2
+
=
𝑥 − 2 𝑥 + 2 𝑥2 − 4
8)
7
6
−5= 2
𝑥−1
𝑥 −1
𝑥 = −3
3
𝑥 = − ,𝑥 = 2
5
1
**9) Consider the function 𝑓 𝑥 = − 2 . Determine whether the inverse of 𝑓 is a
𝑥
function then find the inverse.
𝑦𝑒𝑠, 𝑡ℎ𝑒 𝑖𝑛𝑣𝑒𝑠𝑒 𝑖𝑠 𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛.
𝑔 𝑥 =
1
𝑥+2