Rational Equations and Inequalities PDF - Grade 11

Republic of the Philippines Department of Education REGION II – CAGAYAN VALLEY SCHOOLS DIVISION OF QUIRINO...

Republic of the Philippines Department of Education REGION II – CAGAYAN VALLEY SCHOOLS DIVISION OF QUIRINO SAGUDAY NATIONAL HIGH SCHOOL (INTEGRATED JUNIOR & SENIOR) RATIONAL EQUATIONS AND INEQUALITIES 𝑨 RATIONAL EXPRESSION- a rational expression can be written in the form 𝑩 where a and b ≠ 0. 𝟕 𝟕−𝒙 ex. 𝒂𝒃 , 𝟐𝒙𝟐 +𝟑 RATIONAL EQUATION- is an equation that contains one or more rational expressions. 𝟕 ex. 𝒂𝒃 =4 𝟕−𝒙 =𝒙 𝟐𝒙𝟐 +𝟑 RATIONAL INEQUALITY-is composed of rational expressions combined with 𝒂 ≤, ≥, 𝒔𝒊𝒈𝒏. 𝟕 ex. 𝒂𝒃>b 𝟕−𝒙 ≥ 𝟐𝒙 𝟐𝒙𝟐 +𝟑 RATIONAL FUNCTION Polynomial function is a function defined by 𝑓(𝑥) = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 …+𝑎0 , where 𝑎0, 𝑎1, 𝑎2, … 𝑎𝑛 are real numbers, 𝑎𝑛 ≠ 0, and n is a non-negative integer. A linear function f is a constant function if 𝑓(𝑥) = 𝑚𝑥 + 𝑏, where 𝑚 = 0 and b are any real number. A linear function f is a constant function if 𝑓(𝑥) = 𝑚𝑥 + 𝑏, where 𝑚 and b are any real number. And 𝑚 𝑎𝑛𝑑 𝑓(𝑥) are both not equal to zero. A quadratic function is any equation of the form 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 where a, b and c are real numbers and a ≠ 0. 𝑁(𝑥) A rational function can be written in the form if 𝑓(𝑥) = 𝐷(𝑥) where N(x) and D(x) are polynomials and D(x) is not the zero polynomial. 𝑁(𝑥) The domain of a rational function if 𝑓(𝑥) = 𝐷(𝑥) is all the values of x that will not make the denominator equal to zero. School: SAGUDAY NATIONAL HIGH SCHOOL Address: Magsaysay, Saguday, Quirino Contact Numbers: 078 374 5807 / 0917 125 8579 Email Address: [email protected]  Republic of the Philippines Department of Education REGION II – CAGAYAN VALLEY SCHOOLS DIVISION OF QUIRINO SAGUDAY NATIONAL HIGH SCHOOL (INTEGRATED JUNIOR & SENIOR) 𝑁(𝑥) The domain of a rational function if 𝑓(𝑥) = 𝐷(𝑥) is all the values of x that will not make the denominator equal to zero. Give the domain of the following functions. 2𝑥+5 𝑥−5 𝑥+3 Example: a. 𝑓(𝑥) = b. 𝑓(𝑥) = 𝑥+3 c. 𝑓(𝑥) = 𝑥 2 −9 𝑥−6 Solution: 2𝑥+5 𝑥−5 𝑥+3 To find the a. 𝑓(𝑥) = b. 𝑓(𝑥) = 𝑥+3 c. 𝑓(𝑥) = 𝑥 2 −9 𝑥−6 domain find the 𝑥−6=0 𝑥+3 =0 𝑥2 − 9 = 0 restriction of x. 𝑥=6 𝑥 = −3 𝑥2 = 9 To get the restriction get The restricted x- The restricted x-value of √𝑥 2 = √9 the value of f is 6. f is -3. Hence, the he denominator Hence, the domain of domain of f is the set of 𝑥 = ±3 and find the f is the set of real real numbers except -3. The restricted x-value of f is±3. Hence, the zero. numbers except 6. he domain of f is the set of real numbers except ±3. Activity: 1 whole sheet of paper, find the domain of the following functions. Show your solution. 𝑥−5 𝑥+10 10𝑥−100 1. 𝑓(𝑥) = 𝑥+2 6. 𝑓(𝑥) = 11. 𝑓(𝑥) = 𝑥+4 𝑥−10 𝑥 2 +12𝑥+36 3𝑥−5 𝑥 2 +3𝑥−10 2. 𝑓(𝑥) = 7. 𝑓(𝑥) = 12. 𝑓(𝑥) = 𝑥 2 +5𝑥+6 𝑥+6 𝑥 2 −25 𝑥 2 +2𝑥−1 3𝑥−10 𝑥 2 +8 3. 𝑓(𝑥) = 8. 𝑓(𝑥) = 13.. 𝑓(𝑥) = 𝑥 2−14𝑥+13 𝑥 2 +4𝑥+4 2𝑥+6 𝑥 2 +4𝑥+4 4𝑥−10 3𝑥−12 4. 𝑓(𝑥) = 9. 𝑓(𝑥) = 3𝑥+24 14.. 𝑓(𝑥) = 𝑥 2−𝑥+12 2𝑥+2 𝑥−5 𝑥 2 −10 𝑥 2 −100 5. 𝑓(𝑥) = 10. 𝑓(𝑥) = 3𝑥−15 15.. 𝑓(𝑥) = 𝑥 2+24𝑥−144 𝑥 2 −1 Prepared by: JAMALIA R. HAGI AMEN ********Deadline: August 23, 2024 School: SAGUDAY NATIONAL HIGH SCHOOL Address: Magsaysay, Saguday, Quirino Contact Numbers: 078 374 5807 / 0917 125 8579 Email Address: [email protected]