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 :

                     x/2+13/10-(x/5+1)≤0 

Step by step solution :

Step  1  :

            x
 Simplify   —
            5

Equation at the end of step  1  :

   x    13      x    
  (— +  ——) -  (— +  1)  ≤ 0 
   2    10      5    

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  5  as the denominator :

         1     1 • 5
    1 =  —  =  —————
         1       5  

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:

 x + 5     x + 5
 —————  =  —————
   5         5  

Equation at the end of step  2  :

   x    13     (x + 5)
  (— +  ——) -  ———————  ≤ 0 
   2    10        5   

Step  3  :

            13
 Simplify   ——
            10

Equation at the end of step  3  :

   x    13     (x + 5)
  (— +  ——) -  ———————  ≤ 0 
   2    10        5   

Step  4  :

            x
 Simplify   —
            2

Equation at the end of step  4  :

   x    13     (x + 5)
  (— +  ——) -  ———————  ≤ 0 
   2    10        5   

Step  5  :

Calculating the Least Common Multiple :

 5.1    Find the Least Common Multiple

      The left denominator is :       2 

      The right denominator is :       10 

      Least Common Multiple:
      10 

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 = 5

   Right_M = L.C.M / R_Deno = 1

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.      x • 5
   ——————————————————  =   —————
         L.C.M              10  

   R. Mult. • R. Num.      13
   ——————————————————  =   ——
         L.C.M             10

Adding fractions that have a common denominator :

 5.4       Adding up the two equivalent fractions

 x • 5 + 13     5x + 13
 ——————————  =  ———————
     10           10   

Equation at the end of step  5  :

  (5x + 13)    (x + 5)
  ————————— -  ———————  ≤ 0 
     10           5   

Step  6  :

Calculating the Least Common Multiple :

 6.1    Find the Least Common Multiple

      The left denominator is :       10 

      The right denominator is :       5 

      Least Common Multiple:
      10 

Calculating Multipliers :

 6.2    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 = 1

   Right_M = L.C.M / R_Deno = 2

Making Equivalent Fractions :

 6.3      Rewrite the two fractions into equivalent fractions

   L. Mult. • L. Num.      (5x+13)
   ——————————————————  =   ———————
         L.C.M               10   

   R. Mult. • R. Num.      (x+5) • 2
   ——————————————————  =   —————————
         L.C.M                10    

Adding fractions that have a common denominator :

 6.4       Adding up the two equivalent fractions

 (5x+13) - ((x+5) • 2)     3x + 3
 —————————————————————  =  ——————
          10                 10  

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   3x + 3  =   3 • (x + 1) 

Equation at the end of step  7  :

  3 • (x + 1)
  ———————————  ≤ 0 
      10     

Step  8  :

 8.1    Multiply both sides by  10 

 8.2    Divide both sides by  3 

Solve Basic Inequality :

 8.3      Subtract  1  from both sides

            x ≤ -1

Inequality Plot :

 8.4      Inequality plot for

         0.300 X  + 0.300  ≤  0

  
 

One solution was found :