Solution - Adding, subtracting and finding the least common multiple
Step by Step Solution
Rearrange:
Rearrange the equation by subtracting what is to the right of the greater than sign from both sides of the inequality :
m/3-5-(8/3-m/5)<0
Step by step solution :
Step 1 :
m
Simplify —
5
Equation at the end of step 1 :
m 8 m
(— - 5) - (— - —) < 0
3 3 5
Step 2 :
8
Simplify —
3
Equation at the end of step 2 :
m 8 m
(— - 5) - (— - —) < 0
3 3 5
Step 3 :
Calculating the Least Common Multiple :
3.1 Find the Least Common Multiple
The left denominator is : 3
The right denominator is : 5
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 3 | 1 | 0 | 1 |
| 5 | 0 | 1 | 1 |
| Product of all Prime Factors | 3 | 5 | 15 |
Least Common Multiple:
15
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 = 5
Right_M = L.C.M / R_Deno = 3
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. 8 • 5 —————————————————— = ————— L.C.M 15 R. Mult. • R. Num. m • 3 —————————————————— = ————— L.C.M 15
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:
8 • 5 - (m • 3) 40 - 3m
——————————————— = ———————
15 15
Equation at the end of step 3 :
m (40 - 3m)
(— - 5) - ————————— < 0
3 15
Step 4 :
m
Simplify —
3
Equation at the end of step 4 :
m (40 - 3m)
(— - 5) - ————————— < 0
3 15
Step 5 :
Rewriting the whole as an Equivalent Fraction :
5.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using 3 as the denominator :
5 5 • 3
5 = — = —————
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 :
5.2 Adding up the two equivalent fractions
m - (5 • 3) m - 15
——————————— = ——————
3 3
Equation at the end of step 5 :
(m - 15) (40 - 3m)
———————— - ————————— < 0
3 15
Step 6 :
Calculating the Least Common Multiple :
6.1 Find the Least Common Multiple
The left denominator is : 3
The right denominator is : 15
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 3 | 1 | 1 | 1 |
| 5 | 0 | 1 | 1 |
| Product of all Prime Factors | 3 | 15 | 15 |
Least Common Multiple:
15
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 = 5
Right_M = L.C.M / R_Deno = 1
Making Equivalent Fractions :
6.3 Rewrite the two fractions into equivalent fractions
L. Mult. • L. Num. (m-15) • 5 —————————————————— = —————————— L.C.M 15 R. Mult. • R. Num. (40-3m) —————————————————— = ——————— L.C.M 15
Adding fractions that have a common denominator :
6.4 Adding up the two equivalent fractions
(m-15) • 5 - ((40-3m)) 8m - 115
—————————————————————— = ————————
15 15
Equation at the end of step 6 :
8m - 115
———————— < 0
15
Step 7 :
7.1 Multiply both sides by 15
7.2 Divide both sides by 8
m-(115/8) < 0
Solve Basic Inequality :
7.3 Add 115/8 to both sides
m < 115/8
Inequality Plot :
7.4 Inequality plot for
0.533 m - 7.667 < 0
One solution was found :
m < 115/8How did we do?
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