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 less equal sign from both sides of the inequality :
5/3*x+1/2-(1/3)≤0
Step by step solution :
Step 1 :
1
Simplify —
3
Equation at the end of step 1 :
5 1 1
((— • x) + —) - — ≤ 0
3 2 3
Step 2 :
1
Simplify —
2
Equation at the end of step 2 :
5 1 1
((— • x) + —) - — ≤ 0
3 2 3
Step 3 :
5
Simplify —
3
Equation at the end of step 3 :
5 1 1
((— • x) + —) - — ≤ 0
3 2 3
Step 4 :
Calculating the Least Common Multiple :
4.1 Find the Least Common Multiple
The left denominator is : 3
The right denominator is : 2
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 3 | 1 | 0 | 1 |
| 2 | 0 | 1 | 1 |
| Product of all Prime Factors | 3 | 2 | 6 |
Least Common Multiple:
6
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 = 2
Right_M = L.C.M / R_Deno = 3
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. 5x • 2 —————————————————— = —————— L.C.M 6 R. Mult. • R. Num. 3 —————————————————— = — L.C.M 6
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:
5x • 2 + 3 10x + 3
—————————— = ———————
6 6
Equation at the end of step 4 :
(10x + 3) 1
————————— - — ≤ 0
6 3
Step 5 :
Calculating the Least Common Multiple :
5.1 Find the Least Common Multiple
The left denominator is : 6
The right denominator is : 3
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 2 | 1 | 0 | 1 |
| 3 | 1 | 1 | 1 |
| Product of all Prime Factors | 6 | 3 | 6 |
Least Common Multiple:
6
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 = 2
Making Equivalent Fractions :
5.3 Rewrite the two fractions into equivalent fractions
L. Mult. • L. Num. (10x+3) —————————————————— = ——————— L.C.M 6 R. Mult. • R. Num. 2 —————————————————— = — L.C.M 6
Adding fractions that have a common denominator :
5.4 Adding up the two equivalent fractions
(10x+3) - (2) 10x + 1
————————————— = ———————
6 6
Equation at the end of step 5 :
10x + 1
——————— ≤ 0
6
Step 6 :
6.1 Multiply both sides by 6
6.2 Divide both sides by 10
x+(1/10) ≤ 0
Solve Basic Inequality :
6.3 Subtract 1/10 from both sides
x ≤ -1/10
Inequality Plot :
6.4 Inequality plot for
1.667 x + 0.167 ≤ 0
One solution was found :
x ≤ -1/10How did we do?
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