BPM MATH0 – Week 3
Rational Expressions and Equations
Week 3 – Rational Expressions and Equations
Weekly Goals
ˆ Understand domain restrictions for rational expressions.
ˆ Simplify and manipulate rational expressions.
ˆ Solve rational equations and inequalities.
Solved Examples – With Detailed Steps
Example 1: Simplify:
2
x
−
1
x + 2
+
x + 1
x(x + 2)
Steps:
ˆ Common denominator: x(x + 2)
ˆ Rewrite each term:
2(x + 2)
x(x + 2)
−
1 · x
x(x + 2)
+
x + 1
x(x + 2)
ˆ Expand and combine numerators:
2x + 4 − x + x + 1
x(x + 2)
=
2x + 5
x(x + 2)
2x + 5
x(x + 2)
Example 2: Solve the equation:
x − 3
x + 1
= 2
Steps:
ˆ Multiply both sides by x + 1: x − 3 = 2(x + 1)
ˆ Expand right-hand side: x − 3 = 2x + 2
ˆ Rearranged: −x = 5 ⇒ x = −5
x = −5
BPM MATH0 – Week 3
Rational Expressions and Equations
Example 3: Solve the inequality:
x − 2
x + 4
≤ 1
Steps:
ˆ Move all terms to one side:
x − 2
x + 4
− 1 ≤ 0 ⇒
x − 2 − (x + 4)
x + 4
≤ 0 ⇒
−6
x + 4
≤ 0
ˆ The sign of the expression depends on the denominator: solution is where x+4 > 0
x > −4
Example 4: Find the domain of:
f(x) =
x2
− 1
x2 − 4x + 3
Steps:
ˆ Denominator: x2
− 4x + 3 = (x − 1)(x − 3)
ˆ Expression is undefined for x = 1 and x = 3
Domain: R \ {1, 3}
Practice Problems for Seminar
Simplify
1. 1
x−1
+ 2
x+1
· x2−1
3
2. x2−16
x2+3x−18
· x+6
x−4
3. 3x
x2−1
− 2
x−1
4. 1
x
+ 2
x+1
− x+3
x(x+1)
Solve
5. x+5
x−1
= 2
6. 2x−3
x+2
≤ 1
7. 4
x+1
+ 1
x−1
= 5x
x2−1