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 :

                     1/2+a-(-2/3)<0 

Step by step solution :

Step  1  :

            2
 Simplify   —
            3

Equation at the end of step  1  :

   1                2
  (— +  a) -  (0 -  —)  < 0 
   2                3

Step  2  :

            1
 Simplify   —
            2

Equation at the end of step  2  :

   1          -2
  (— +  a) -  ——  < 0 
   2          3 

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a whole to a fraction

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

         a     a • 2
    a =  —  =  —————
         1       2  

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 :

 3.2       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:

 1 + a • 2     2a + 1
 —————————  =  ——————
     2           2   

Equation at the end of step  3  :

  (2a + 1)    -2
  ———————— -  ——  < 0 
     2        3 

Step  4  :

Calculating the Least Common Multiple :

 4.1    Find the Least Common Multiple

      The left denominator is :       2 

      The right denominator is :       3 

      Least Common Multiple:
      6 

Calculating Multipliers :

 4.2    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 = 3

   Right_M = L.C.M / R_Deno = 2

Making Equivalent Fractions :

 4.3      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.      (2a+1) • 3
   ——————————————————  =   ——————————
         L.C.M                 6     

   R. Mult. • R. Num.      -2 • 2
   ——————————————————  =   ——————
         L.C.M               6   

Adding fractions that have a common denominator :

 4.4       Adding up the two equivalent fractions

 (2a+1) • 3 - (-2 • 2)     6a + 7
 —————————————————————  =  ——————
           6                 6   

Equation at the end of step  4  :

  6a + 7
  ——————  < 0 
    6   

Step  5  :

 5.1    Multiply both sides by  6 

 5.2    Divide both sides by  6  

      a+(7/6)  < 0

Solve Basic Inequality :

 5.3      Subtract  7/6  from both sides

            a < -7/6

Inequality Plot :

 5.4      Inequality plot for

         a  + 1.167  <  0

  
 

One solution was found :