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Solution - Adding, subtracting and finding the least common multiple

x \geq \frac{27}{2}

x>=27/2

Step by Step Solution

Step by step solution :

Step 1 :

            -20x
 Simplify   ————
             16

Equation at the end of step 1 :

  270    -5x
  ——— +  ———  ≤ 0 
  16      4

Step 2 :

            135
 Simplify   ———
             8

Equation at the end of step 2 :

  135    -5x
  ——— +  ———  ≤ 0 
   8      4

Step 3 :

Calculating the Least Common Multiple :

3.1    Find the Least Common Multiple

      The left denominator is :       8

      The right denominator is :       4

Number of times each prime factor
appears in the factorization of:
Prime Factor	Left Denominator	Right Denominator	L.C.M = Max {Left,Right}
2	3	2	3
Product of all Prime Factors	8	4	8

Least Common Multiple:
8

Calculating Multipliers :

3.2    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 :

3.3 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.      135
   ——————————————————  =   ———
         L.C.M              8 

   R. Mult. • R. Num.      -5x • 2
   ——————————————————  =   ———————
         L.C.M                8

Adding fractions that have a common denominator :

3.4 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:

 135 + -5x • 2     135 - 10x
 —————————————  =  —————————
       8               8

Step 4 :

Pulling out like terms :

4.1 Pull out like factors :

135 - 10x = -5 • (2x - 27)

Equation at the end of step 4 :

  -5 • (2x - 27)
  ——————————————  ≤ 0 
        8

Step 5 :

5.1    Multiply both sides by 8

5.2    Divide both sides by -5

Remember to flip the inequality sign:

5.3    Divide both sides by 2

      x-(27/2) ≥ 0

Solve Basic Inequality :

 5.4      Add  27/2  to both sides

             x ≥ 27/2

Inequality Plot :

 5.5      Inequality plot for

         -1.250 X  + 16.875  ≥  0

One solution was found :

x ≥ 27/2

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