Solution - Adding, subtracting and finding the least common multiple

Rearrange:

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

                     3/4*x-(-3/2*x-6)≥0 

Step by step solution :

Step  1  :

            3
 Simplify   —
            2

Equation at the end of step  1  :

   3         3
  (—•x)-((0-(—•x))-6)  ≥ 0 
   4         2

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a whole from a fraction

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

         6     6 • 2
    6 =  —  =  —————
         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 :

 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:

 -3x - (6 • 2)     -3x - 12
 —————————————  =  ————————
       2              2    

Equation at the end of step  2  :

   3         (-3x - 12)
  (— • x) -  ——————————  ≥ 0 
   4             2     

Step  3  :

            3
 Simplify   —
            4

Equation at the end of step  3  :

   3         (-3x - 12)
  (— • x) -  ——————————  ≥ 0 
   4             2     

Step  4  :

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   -3x - 12  =   -3 • (x + 4) 

Calculating the Least Common Multiple :

 5.2    Find the Least Common Multiple

      The left denominator is :       4 

      The right denominator is :       2 

      Least Common Multiple:
      4 

Calculating Multipliers :

 5.3    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 = 1

   Right_M = L.C.M / R_Deno = 2

Making Equivalent Fractions :

 5.4      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.      3x
   ——————————————————  =   ——
         L.C.M             4 

   R. Mult. • R. Num.      -3 • (x+4) • 2
   ——————————————————  =   ——————————————
         L.C.M                   4       

Adding fractions that have a common denominator :

 5.5       Adding up the two equivalent fractions

 3x - (-3 • (x+4) • 2)     9x + 24
 —————————————————————  =  ———————
           4                  4   

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   9x + 24  =   3 • (3x + 8) 

Equation at the end of step  6  :

  3 • (3x + 8)
  ————————————  ≥ 0 
       4      

Step  7  :

 7.1    Multiply both sides by  4 

 7.2    Divide both sides by  3 

 7.3    Divide both sides by  3  

      x+(8/3)  ≥ 0

Solve Basic Inequality :

 7.4      Subtract  8/3  from both sides

            x ≥ -8/3

Inequality Plot :

 7.5      Inequality plot for

         2.250 X  + 6.000  ≥  0

  
 

One solution was found :