What is the sum of the rational expressions below? x/2 + 7/9x

Accepted Solution

Answer:[tex]\frac{9\,x^2+14}{18\,x}[/tex]Step-by-step explanation:Recall that fractions can be combined ONLY if they have the SAME denominator. Therefore, in order to combine the fractions given, we need to first write them with a same common denominator. For such, we study the factors that each denominator has, and use them to create the Greatest Common Factor of the two denominators. That is going to be the common denominator we need to use in order to express our fractions and be able to combine them.The first fraction ([tex]\frac{x}{2}[/tex]) has only the factor "2" in the denominator, so we need to include it in our collection of greatest common factors.The second fraction ([tex]\frac{7}{9x}[/tex]) has the factors: [tex]3^2 and x[/tex], which we need to include in our greatest common factor.Therefore our greatest common factor consists of the product:[tex]2\,*\,3^2\,*\,x=18\,x[/tex]That means that we need to re-write our original fractions with this denominator. We do such by multiplying both, numerator and denominator of each rational expression by the appropriate factors that would generate the denominator "18 x":To obtain such, we need to multiply numerator and denominator of the first fraction ([tex]\frac{x}{2}[/tex]) by: "9 x" (leading to our goal of getting "18 x" in the denominator):[tex]\frac{x*9\,x}{2*9\, x} =\frac{9\,x^2}{18\,x}[/tex]Now, we need to multiply numerator and denominator of the second fraction ([tex]\frac{7}{9x}[/tex]) by: "2" (leading to our goal of getting "18 x" in the denominator):[tex]\frac{7\,*\,2}{9\, x\,*\,2} =\frac{14}{18\,x}[/tex]So, now our fractions can be combined by direct addition of their numerators: [tex]\frac{9\,x^2}{18\,x}+\frac{14}{18\,x}=\frac{9\,x^2+14}{18\,x}[/tex]