Solution - Adding, subtracting and finding the least common multiple

Rearrange:

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

                     1/2*m+2/3-(2/3*m+2)≤0 

Step by step solution :

Step  1  :

            2
 Simplify   —
            3

Equation at the end of step  1  :

    1    2    2
  ((—•m)+—)-((—•m)+2)  ≤ 0 
    2    3    3

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  3  as the denominator :

         2     2 • 3
    2 =  —  =  —————
         1       3  

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:

 2m + 2 • 3     2m + 6
 ——————————  =  ——————
     3            3   

Equation at the end of step  2  :

    1         2     (2m + 6)
  ((— • m) +  —) -  ————————  ≤ 0 
    2         3        3    

Step  3  :

            2
 Simplify   —
            3

Equation at the end of step  3  :

    1         2     (2m + 6)
  ((— • m) +  —) -  ————————  ≤ 0 
    2         3        3    

Step  4  :

            1
 Simplify   —
            2

Equation at the end of step  4  :

    1         2     (2m + 6)
  ((— • m) +  —) -  ————————  ≤ 0 
    2         3        3    

Step  5  :

Calculating the Least Common Multiple :

 5.1    Find the Least Common Multiple

      The left denominator is :       2 

      The right denominator is :       3 

      Least Common Multiple:
      6 

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 = 3

   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.      m • 3
   ——————————————————  =   —————
         L.C.M               6  

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

Adding fractions that have a common denominator :

 5.4       Adding up the two equivalent fractions

 m • 3 + 2 • 2     3m + 4
 —————————————  =  ——————
       6             6   

Equation at the end of step  5  :

  (3m + 4)    (2m + 6)
  ———————— -  ————————  ≤ 0 
     6           3    

Step  6  :

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   2m + 6  =   2 • (m + 3) 

Calculating the Least Common Multiple :

 7.2    Find the Least Common Multiple

      The left denominator is :       6 

      The right denominator is :       3 

      Least Common Multiple:
      6 

Calculating Multipliers :

 7.3    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 :

 7.4      Rewrite the two fractions into equivalent fractions

   L. Mult. • L. Num.      (3m+4)
   ——————————————————  =   ——————
         L.C.M               6   

   R. Mult. • R. Num.      2 • (m+3) • 2
   ——————————————————  =   —————————————
         L.C.M                   6      

Adding fractions that have a common denominator :

 7.5       Adding up the two equivalent fractions

 (3m+4) - (2 • (m+3) • 2)     -m - 8
 ————————————————————————  =  ——————
            6                   6   

Step  8  :

Pulling out like terms :

 8.1     Pull out like factors :

   -m - 8  =   -1 • (m + 8) 

Equation at the end of step  8  :

  -m - 8
  ——————  ≤ 0 
    6   

Step  9  :

 9.1    Multiply both sides by  6 

 9.2    Multiply both sides by (-1)

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

      m+8  ≥ 0

Solve Basic Inequality :

 9.3      Subtract  8  from both sides

            m ≥ -8

Inequality Plot :

 9.4      Inequality plot for

         -0.167 m  - 1.333  ≤  0

  
 

One solution was found :