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): "8.2" was replaced by "(82/10)". 3 more similar replacement(s)
Rearrange:
Rearrange the equation by subtracting what is to the right of the less equal sign from both sides of the inequality :
(65/10)*x-(113/10)-((82/10))≤0
Step by step solution :
Step 1 :
41
Simplify ——
5
Equation at the end of step 1 :
65 113 41
((—— • x) - ———) - —— ≤ 0
10 10 5
Step 2 :
113
Simplify ———
10
Equation at the end of step 2 :
65 113 41
((—— • x) - ———) - —— ≤ 0
10 10 5
Step 3 :
13
Simplify ——
2
Equation at the end of step 3 :
13 113 41
((—— • x) - ———) - —— ≤ 0
2 10 5
Step 4 :
Calculating the Least Common Multiple :
4.1 Find the Least Common Multiple
The left denominator is : 2
The right denominator is : 10
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 2 | 1 | 1 | 1 |
| 5 | 0 | 1 | 1 |
| Product of all Prime Factors | 2 | 10 | 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 = 5
Right_M = L.C.M / R_Deno = 1
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. 13x • 5 —————————————————— = ——————— L.C.M 10 R. Mult. • R. Num. 113 —————————————————— = ——— 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:
13x • 5 - (113) 65x - 113
——————————————— = —————————
10 10
Equation at the end of step 4 :
(65x - 113) 41
——————————— - —— ≤ 0
10 5
Step 5 :
Calculating the Least Common Multiple :
5.1 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 :
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. (65x-113) —————————————————— = ————————— L.C.M 10 R. Mult. • R. Num. 41 • 2 —————————————————— = —————— L.C.M 10
Adding fractions that have a common denominator :
5.4 Adding up the two equivalent fractions
(65x-113) - (41 • 2) 65x - 195
———————————————————— = —————————
10 10
Step 6 :
Pulling out like terms :
6.1 Pull out like factors :
65x - 195 = 65 • (x - 3)
Equation at the end of step 6 :
65 • (x - 3)
———————————— ≤ 0
10
Step 7 :
7.1 Multiply both sides by 10
7.2 Divide both sides by 65
Solve Basic Inequality :
7.3 Add 3 to both sides
x ≤ 3
Inequality Plot :
7.4 Inequality plot for
6.500 X - 19.500 ≤ 0
One solution was found :
x ≤ 3How did we do?
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