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 :
3/2*p-2/3-(4/9+1/2*p)<0
Step by step solution :
Step 1 :
1
Simplify —
2
Equation at the end of step 1 :
3 2 4 1
((—•p)-—)-(—+(—•p)) < 0
2 3 9 2
Step 2 :
4
Simplify —
9
Equation at the end of step 2 :
3 2 4 p
((—•p)-—)-(—+—) < 0
2 3 9 2
Step 3 :
Calculating the Least Common Multiple :
3.1 Find the Least Common Multiple
The left denominator is : 9
The right denominator is : 2
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 3 | 2 | 0 | 2 |
| 2 | 0 | 1 | 1 |
| Product of all Prime Factors | 9 | 2 | 18 |
Least Common Multiple:
18
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 = 2
Right_M = L.C.M / R_Deno = 9
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. 4 • 2 —————————————————— = ————— L.C.M 18 R. Mult. • R. Num. p • 9 —————————————————— = ————— L.C.M 18
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:
4 • 2 + p • 9 9p + 8
————————————— = ——————
18 18
Equation at the end of step 3 :
3 2 (9p + 8)
((— • p) - —) - ———————— < 0
2 3 18
Step 4 :
2
Simplify —
3
Equation at the end of step 4 :
3 2 (9p + 8)
((— • p) - —) - ———————— < 0
2 3 18
Step 5 :
3
Simplify —
2
Equation at the end of step 5 :
3 2 (9p + 8)
((— • p) - —) - ———————— < 0
2 3 18
Step 6 :
Calculating the Least Common Multiple :
6.1 Find the Least Common Multiple
The left denominator is : 2
The right denominator is : 3
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 2 | 1 | 0 | 1 |
| 3 | 0 | 1 | 1 |
| Product of all Prime Factors | 2 | 3 | 6 |
Least Common Multiple:
6
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 = 3
Right_M = L.C.M / R_Deno = 2
Making Equivalent Fractions :
6.3 Rewrite the two fractions into equivalent fractions
L. Mult. • L. Num. 3p • 3 —————————————————— = —————— L.C.M 6 R. Mult. • R. Num. 2 • 2 —————————————————— = ————— L.C.M 6
Adding fractions that have a common denominator :
6.4 Adding up the two equivalent fractions
3p • 3 - (2 • 2) 9p - 4
———————————————— = ——————
6 6
Equation at the end of step 6 :
(9p - 4) (9p + 8)
———————— - ———————— < 0
6 18
Step 7 :
Calculating the Least Common Multiple :
7.1 Find the Least Common Multiple
The left denominator is : 6
The right denominator is : 18
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 2 | 1 | 1 | 1 |
| 3 | 1 | 2 | 2 |
| Product of all Prime Factors | 6 | 18 | 18 |
Least Common Multiple:
18
Calculating Multipliers :
7.2 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 = 3
Right_M = L.C.M / R_Deno = 1
Making Equivalent Fractions :
7.3 Rewrite the two fractions into equivalent fractions
L. Mult. • L. Num. (9p-4) • 3 —————————————————— = —————————— L.C.M 18 R. Mult. • R. Num. (9p+8) —————————————————— = —————— L.C.M 18
Adding fractions that have a common denominator :
7.4 Adding up the two equivalent fractions
(9p-4) • 3 - ((9p+8)) 18p - 20
————————————————————— = ————————
18 18
Step 8 :
Pulling out like terms :
8.1 Pull out like factors :
18p - 20 = 2 • (9p - 10)
Equation at the end of step 8 :
2 • (9p - 10)
————————————— < 0
18
Step 9 :
9.1 Multiply both sides by 18
9.2 Divide both sides by 2
9.3 Divide both sides by 9
p-(10/9) < 0
Solve Basic Inequality :
9.4 Add 10/9 to both sides
p < 10/9
Inequality Plot :
9.5 Inequality plot for
X - 1.111 < 0
One solution was found :
p < 10/9How did we do?
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