Solution - Adding, subtracting and finding the least common multiple
Step by Step Solution
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "74.5" was replaced by "(745/10)". 3 more similar replacement(s)
Rearrange:
Rearrange the equation by subtracting what is to the right of the greater than sign from both sides of the inequality :
(493/10)+(48/10)*r-((745/10))<0
Step by step solution :
Step 1 :
149
Simplify ———
2
Equation at the end of step 1 :
493 48 149
(——— + (—— • r)) - ——— < 0
10 10 2
Step 2 :
24
Simplify ——
5
Equation at the end of step 2 :
493 24 149
(——— + (—— • r)) - ——— < 0
10 5 2
Step 3 :
493
Simplify ———
10
Equation at the end of step 3 :
493 24r 149
(——— + ———) - ——— < 0
10 5 2
Step 4 :
Calculating the Least Common Multiple :
4.1 Find the Least Common Multiple
The left denominator is : 10
The right denominator is : 5
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 2 | 1 | 0 | 1 |
| 5 | 1 | 1 | 1 |
| Product of all Prime Factors | 10 | 5 | 10 |
Least Common Multiple:
10
Calculating Multipliers :
4.2 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 :
4.3 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. 493 —————————————————— = ——— L.C.M 10 R. Mult. • R. Num. 24r • 2 —————————————————— = ——————— L.C.M 10
Adding fractions that have a common denominator :
4.4 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:
493 + 24r • 2 48r + 493
————————————— = —————————
10 10
Equation at the end of step 4 :
(48r + 493) 149
——————————— - ——— < 0
10 2
Step 5 :
Calculating the Least Common Multiple :
5.1 Find the Least Common Multiple
The left denominator is : 10
The right denominator is : 2
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 2 | 1 | 1 | 1 |
| 5 | 1 | 0 | 1 |
| Product of all Prime Factors | 10 | 2 | 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 = 1
Right_M = L.C.M / R_Deno = 5
Making Equivalent Fractions :
5.3 Rewrite the two fractions into equivalent fractions
L. Mult. • L. Num. (48r+493) —————————————————— = ————————— L.C.M 10 R. Mult. • R. Num. 149 • 5 —————————————————— = ——————— L.C.M 10
Adding fractions that have a common denominator :
5.4 Adding up the two equivalent fractions
(48r+493) - (149 • 5) 48r - 252
————————————————————— = —————————
10 10
Step 6 :
Pulling out like terms :
6.1 Pull out like factors :
48r - 252 = 12 • (4r - 21)
Equation at the end of step 6 :
12 • (4r - 21)
—————————————— < 0
10
Step 7 :
7.1 Multiply both sides by 10
7.2 Divide both sides by 12
7.3 Divide both sides by 4
r-(21/4) < 0
Solve Basic Inequality :
7.4 Add 21/4 to both sides
r < 21/4
Inequality Plot :
7.5 Inequality plot for
4.800 X - 25.200 < 0
One solution was found :
r < 21/4How did we do?
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