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 :

                     z+2/11-(8*z-3/4)>0 

Step by step solution :

Step  1  :

            3
 Simplify   —
            4

Equation at the end of step  1  :

         2            3
  (z +  ——) -  (8z -  —)  > 0 
        11            4

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  4  as the denominator :

           8z     8z • 4
     8z =  ——  =  ——————
           1        4   

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:

 8z • 4 - (3)     32z - 3
 ————————————  =  ———————
      4              4   

Equation at the end of step  2  :

         2     (32z - 3)
  (z +  ——) -  —————————  > 0 
        11         4    

Step  3  :

             2
 Simplify   ——
            11

Equation at the end of step  3  :

         2     (32z - 3)
  (z +  ——) -  —————————  > 0 
        11         4    

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Adding a fraction to a whole

Rewrite the whole as a fraction using  11  as the denominator :

          z     z • 11
     z =  —  =  ——————
          1       11  

Adding fractions that have a common denominator :

 4.2       Adding up the two equivalent fractions

 z • 11 + 2     11z + 2
 ——————————  =  ———————
     11           11   

Equation at the end of step  4  :

  (11z + 2)    (32z - 3)
  ————————— -  —————————  > 0 
     11            4    

Step  5  :

Calculating the Least Common Multiple :

 5.1    Find the Least Common Multiple

      The left denominator is :       11 

      The right denominator is :       4 

      Least Common Multiple:
      44 

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 = 4

   Right_M = L.C.M / R_Deno = 11

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.      (11z+2) • 4
   ——————————————————  =   ———————————
         L.C.M                 44     

   R. Mult. • R. Num.      (32z-3) • 11
   ——————————————————  =   ————————————
         L.C.M                  44     

Adding fractions that have a common denominator :

 5.4       Adding up the two equivalent fractions

 (11z+2) • 4 - ((32z-3) • 11)     41 - 308z
 ————————————————————————————  =  —————————
              44                     44    

Equation at the end of step  5  :

  41 - 308z
  —————————  > 0 
     44    

Step  6  :

 6.1    Multiply both sides by  44 

 6.2    Multiply both sides by (-1)

Flip the inequality sign since you are multiplying by a negative number

      308z-41  < 0

 6.3    Divide both sides by  308  

      z-(41/308)  < 0

Solve Basic Inequality :

 6.4      Add  41/308  to both sides

             z < 41/308

Inequality Plot :

 6.5      Inequality plot for

         -7.000 z  + 0.932  >  0

  
 

One solution was found :