Solution - Adding, subtracting and finding the least common multiple

Rearrange:

Rearrange the equation by subtracting what is to the right of the greater than sign from both sides of the inequality :

                     2*x+1/2-(2/9*x-1)>0 

Step by step solution :

Step  1  :

            2
 Simplify   —
            9

Equation at the end of step  1  :

         1       2         
  (2x +  —) -  ((— • x) -  1)  > 0 
         2       9         

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  9  as the denominator :

         1     1 • 9
    1 =  —  =  —————
         1       9  

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 2.2       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 2x - (9)     2x - 9
 ————————  =  ——————
    9           9   

Equation at the end of step  2  :

         1     (2x - 9)
  (2x +  —) -  ————————  > 0 
         2        9    

Step  3  :

            1
 Simplify   —
            2

Equation at the end of step  3  :

         1     (2x - 9)
  (2x +  —) -  ————————  > 0 
         2        9    

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Adding a fraction to a whole

Rewrite the whole as a fraction using  2  as the denominator :

           2x     2x • 2
     2x =  ——  =  ——————
           1        2   

Adding fractions that have a common denominator :

 4.2       Adding up the two equivalent fractions

 2x • 2 + 1     4x + 1
 ——————————  =  ——————
     2            2   

Equation at the end of step  4  :

  (4x + 1)    (2x - 9)
  ———————— -  ————————  > 0 
     2           9    

Step  5  :

Calculating the Least Common Multiple :

 5.1    Find the Least Common Multiple

      The left denominator is :       2 

      The right denominator is :       9 

      Least Common Multiple:
      18 

Calculating Multipliers :

 5.2    Calculate multipliers for the two fractions


    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = 9

   Right_M = L.C.M / R_Deno = 2

Making Equivalent Fractions :

 5.3      Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example :  1/2   and  2/4  are equivalent,  y/(y+1)²   and  (y²+y)/(y+1)³  are equivalent as well.

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

   L. Mult. • L. Num.      (4x+1) • 9
   ——————————————————  =   ——————————
         L.C.M                 18    

   R. Mult. • R. Num.      (2x-9) • 2
   ——————————————————  =   ——————————
         L.C.M                 18    

Adding fractions that have a common denominator :

 5.4       Adding up the two equivalent fractions

 (4x+1) • 9 - ((2x-9) • 2)     32x + 27
 —————————————————————————  =  ————————
            18                    18   

Equation at the end of step  5  :

  32x + 27
  ————————  > 0 
     18   

Step  6  :

 6.1    Multiply both sides by  18 

 6.2    Divide both sides by  32  

      x+(27/32)  > 0

Solve Basic Inequality :

 6.3      Subtract  27/32  from both sides

            x > -27/32

Inequality Plot :

 6.4      Inequality plot for

         1.778 x  + 1.500  >  0

  
 

One solution was found :