Hello mga kawawmat! So ang i-discuss natin ngayon ay i-differentiate natin yung pagkakaiba ng rational function, rational equation, and rational inequalities. Bago yan, i-describe muna natin kung ano ba yung rational expression. A rational expression is an expression that can be written as a ratio of two polynomials.
So pag sinabi natin polynomials, so yung mga expression nila dapat walang negative yung exponent, walang fraction yung exponent, at yung variable o yung expression ay wala sa loob ng radical. A rational expression also can be described as a function where either the numerator, denominator, or both have a variable on it. So, kagaya ng mga halimbawa na to is a rational expression. Yung 2 over x, x squared plus 2x plus 3 over x plus 1, 5 over x minus 3. So, both of them are rational expression. So, let's try to have an activity.
Identify the following algebraic expression if they are rational or not. First, x squared plus 3x plus 2 all over x plus 4. So, una muna identify natin kung yung both numerator and denominator ba ay rational or polynomials rather. So, since yung numerator and denominator natin is polynomials, so therefore this is a rational expression. Next, we have 1 over 3x squared. So, yung 1 is up.
polynomial. And all over 3x squared, so it's also a polynomial, so this is a rational expression. Another, we have x squared plus 4x minus 3, all over 2, so kagaya ng halimbawa kanina, so this is also a rational expression.
So we have the square root of x plus 1 all over x cubed minus 1. So tingin natin yung numerator natin. So yung x plus 1 natin ay nasa loob ng radical. So ibig sabihin, ito ay hindi siya polynomial. So therefore, this is not a rational expression. Next, we have x raised to negative 2 minus 5 over x cubed minus 1. So, tingnan natin, may giyong numerator natin.
So, ito ay hindi polynomial. Bakit? Yung exponent dito ay negative.
So, therefore, this is not a rational expression. So, another, we have 1 over x plus 2 over x minus 2. So, kapag sinimplify natin itong fraction na ito, so, this is equivalent to 1 over x plus 2 times x minus 2. So, pag sinimplify natin yan at in-expand natin ito, yung x plus 2 times x minus 2, that is a polynomial. So, therefore, this expression is a rational expression.
So ngayon, alam nyo na yung idea about rational expression na kung saan gagamitin natin para ma-identify natin yung tatlo, yung rational equation, rational inequalities, and rational function. Okay, una yung rational equation. This is an equation involving rational expression.
So, makikita natin, halimbawa, meron tayong 5 over x minus 3 over 2x is equal to 1 over 5. So, madaling tandaan ang rational equation dahil sa word na equation. So, nandyan yung nakakabit yung word na equal. So, tatandaan nyo lang na kapag sinabi natin yung rational equation, So, nakakabit yung salitang equal.
So, yan lang ang tatandaan natin. So, madaling tandaan kung ano yung rational equation. Next, yung rational inequality naman, it involve an inequality involving rational expression.
So kapag sinabi natin inequality, of course, hindi siya equal. Ano ba yung mga simbol na pwede natin gamitin kapag hindi equal yung expression? So kagaya ng halimbawa na to, 5 over x minus 3 is less than or equal to 5 over x minus 3. Equal to 2 over x. So ito yung mga simbol na ginagamit sa inequality.
Ibig sabihin, hindi sila equal. So ano pa yung ibang simbol na pwede natin gamitin? Kapag nakita nyo itong mga simbol na ito sa expression, like greater than or equal, greater than or less than, so yan ay halimbawa ng rational inequality. Rational function, a function of the form f of x is equal to p over x over q of x, where p of x and q of x are polynomial function and q of x is not the zero function, or your q of x is not equal to zero. Halimbawa, f of x is equal to x squared plus 2x plus 3 all over x plus 1. So ano naman yung pagkakaiba ni rational function kay rational equation at rational inequality?
Tatandaan natin yung form na to. Meron siyang f of x at alam natin sa function natin that our f of x is equal to y. So kapag nakita natin in the form of f of x or So In the form of y, ibig sabihin that is a rational function. So yan lang yung pagkakaiba ng tatlo.
Kapag rational equation, tatanda natin merong equal symbol. Kapag rational inequality, merong inequality symbol like greater than, less than, greater than or equal, less than or equal. Kapag rational function naman, in the form of f of x or in the form of.
So let's try to have and practice exercises. So by the way, a rational equation or inequality can be solved for all x values that satisfy the equation or inequality. A rational function expresses a relationship between two variables such as x and y and can be represented by a table of values or a graph.
So, sa mga susunod na lesson natin, ididiscuss natin, mas lalo pa natin i-expand yung bawat isa. Now, let's have first a practice exercises. Tingnan nyo, sagutan nyo sa sarili nyo kung natatandaan yung pagkakaiba ng tatlo.
Determine whether the given rational function is a rational function, a rational equation, a rational inequality, or none of these. First, 2 plus x over x minus 1 is equal to 8. Siyempre, unang titignan kung rational ba yung given. Since rational naman yung given, next yung titignan yung mga simbol na ginagamit. Since makikita nyo may equal symbol yan, so this is a rational equation.
Pangalawa, we have x is greater than the square root of x plus 2. For example, number 2. 2, alam agad natin na hindi siya rational. Bakit? Hindi siya in the form of ratio of 2 polynomials.
At saka, meron tayong radicals. Hindi siya polynomial. So, therefore, this is, our answer is none of these.
So, hindi siya rational equation, hindi siya rational inequality or rational function. Another, we have f of x is equal to 6 minus x plus 3 over x squared minus 5. So, this is an example of rational expression. So, alam natin na this is a rational expression. And, anong next natin titignan?
Yung mga sign or symbol na ginagamit at kung anong form. Since meron tayong f of x, Therefore, yan ay rational function. Next, 2x is greater than or equal to 7 over x plus 4. Again, this is a rational expression.
Ano ang titignan natin symbol? May ginamit siyang greater than or equal. And this is an inequality symbol. So, ibig sabihin yan ay rational inequality.
Last, we have x over 2 is equal to 4 over x plus 9x cubed. So, yung given naman ay rational expression. And then, may ginami tayong symbol na equals and that is rational equation.
So, I hope na differentiate nyo yung tatlo ang pagkakaiba ni rational equation, rational inequality at rational function. Thank you for watching this video. I hope you learned something.
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