SAT-40: Rational Equations and Domain Restrictions

SAT-40: Rational Equations and Domain Restrictions

Description: A rational equation has a variable in the denominator, like 1/x + 1/2 = 3/x. To solve, you clear the fractions — but you must watch for values that make a denominator zero, since those are forbidden (domain restrictions).

The method

1. Note the restricted values: any x that makes a denominator 0 is not allowed.
2. Multiply every term by the LCD (least common denominator) to clear fractions.
3. Solve the resulting equation.
4. Reject any solution that equals a restricted value.

(Oʻzbekcha: maxrajni 0 qiladigan x qiymatlari taqiqlangan — ularni javobdan chiqarib tashlang.)

Why restrictions exist

Division by zero is undefined, so x can never make any denominator 0. Even if your algebra produces such a value, it is not a valid solution.

Rule: list the forbidden x-values before you solve, then cross-check at the end. (Oʻzbekcha: 0 ga boʻlish aniqlanmagan, shuning uchun bunday qiymatlar yechim boʻla olmaydi.)

Worked Example

Solve 1/x + 1/2 = 3/x.

• Restriction: x ≠ 0 (denominator can't be zero).
• LCD = 2x. Multiply each term: 2x·(1/x) + 2x·(1/2) = 2x·(3/x) → 2 + x = 6.
• Solve: x = 4. Since 4 ≠ 0, it is allowed → x = 4.

(Oʻzbekcha: x = 4 ruxsat etilgan, chunki u maxrajni nolga aylantirmaydi.)

Practice

Solve (x)/(x − 2) = 4/(x − 2).

Key words — Kalit soʻzlar

• Rational equation — ratsional (kasrli) tenglama
• Denominator — maxraj
• Numerator — surat
• Domain — aniqlanish sohasi
• Restriction — cheklov (taqiq)
• LCD (least common denominator) — eng kichik umumiy maxraj
• Clear the fractions — kasrlardan qutulish
• Undefined — aniqlanmagan
• Reject a solution — yechimni rad etish

Summary

• List restricted values (denominator = 0) first — they can never be solutions.
• Multiply through by the LCD to clear fractions, then solve.
• Reject any answer that hits a restricted value.