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 :
4/5-(2/3-2/7*x)>0
Step by step solution :
Step 1 :
2
Simplify —
7
Equation at the end of step 1 :
4 2 2
— - (— - (— • x)) > 0
5 3 7
Step 2 :
2
Simplify —
3
Equation at the end of step 2 :
4 2 2x
— - (— - ——) > 0
5 3 7
Step 3 :
Calculating the Least Common Multiple :
3.1 Find the Least Common Multiple
The left denominator is : 3
The right denominator is : 7
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 3 | 1 | 0 | 1 |
| 7 | 0 | 1 | 1 |
| Product of all Prime Factors | 3 | 7 | 21 |
Least Common Multiple:
21
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 = 7
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. 2 • 7 —————————————————— = ————— L.C.M 21 R. Mult. • R. Num. 2x • 3 —————————————————— = —————— L.C.M 21
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:
2 • 7 - (2x • 3) 14 - 6x
———————————————— = ———————
21 21
Equation at the end of step 3 :
4 (14 - 6x)
— - ————————— > 0
5 21
Step 4 :
4
Simplify —
5
Equation at the end of step 4 :
4 (14 - 6x)
— - ————————— > 0
5 21
Step 5 :
Step 6 :
Pulling out like terms :
6.1 Pull out like factors :
14 - 6x = -2 • (3x - 7)
Calculating the Least Common Multiple :
6.2 Find the Least Common Multiple
The left denominator is : 5
The right denominator is : 21
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 5 | 1 | 0 | 1 |
| 3 | 0 | 1 | 1 |
| 7 | 0 | 1 | 1 |
| Product of all Prime Factors | 5 | 21 | 105 |
Least Common Multiple:
105
Calculating Multipliers :
6.3 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 = 21
Right_M = L.C.M / R_Deno = 5
Making Equivalent Fractions :
6.4 Rewrite the two fractions into equivalent fractions
L. Mult. • L. Num. 4 • 21 —————————————————— = —————— L.C.M 105 R. Mult. • R. Num. -2 • (3x-7) • 5 —————————————————— = ——————————————— L.C.M 105
Adding fractions that have a common denominator :
6.5 Adding up the two equivalent fractions
4 • 21 - (-2 • (3x-7) • 5) 30x + 14
—————————————————————————— = ————————
105 105
Step 7 :
Pulling out like terms :
7.1 Pull out like factors :
30x + 14 = 2 • (15x + 7)
Equation at the end of step 7 :
2 • (15x + 7)
————————————— > 0
105
Step 8 :
8.1 Multiply both sides by 105
8.2 Divide both sides by 2
8.3 Divide both sides by 15
x+(7/15) > 0
Solve Basic Inequality :
8.4 Subtract 7/15 from both sides
x > -7/15
Inequality Plot :
8.5 Inequality plot for
0.286 X + 0.133 > 0
One solution was found :
x > -7/15How did we do?
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