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): "0.6" was replaced by "(6/10)". 4 more similar replacement(s)
Rearrange:
Rearrange the equation by subtracting what is to the right of the greater equal sign from both sides of the inequality :
(21/10)*x+(24/10)-((36/10)*x-(6/10))≥0
Step by step solution :
Step 1 :
3
Simplify —
5
Equation at the end of step 1 :
21 24 36 3
((——•x)+——)-((——•x)-—) ≥ 0
10 10 10 5
Step 2 :
18
Simplify ——
5
Equation at the end of step 2 :
21 24 18 3
((——•x)+——)-((——•x)-—) ≥ 0
10 10 5 5
Step 3 :
Adding fractions which have a common denominator :
3.1 Adding fractions which have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
18x - (3) 18x - 3
————————— = ———————
5 5
Equation at the end of step 3 :
21 24 (18x - 3)
((—— • x) + ——) - ————————— ≥ 0
10 10 5
Step 4 :
12
Simplify ——
5
Equation at the end of step 4 :
21 12 (18x - 3)
((—— • x) + ——) - ————————— ≥ 0
10 5 5
Step 5 :
21
Simplify ——
10
Equation at the end of step 5 :
21 12 (18x - 3)
((—— • x) + ——) - ————————— ≥ 0
10 5 5
Step 6 :
Calculating the Least Common Multiple :
6.1 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 :
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 = 1
Right_M = L.C.M / R_Deno = 2
Making Equivalent Fractions :
6.3 Rewrite the two fractions into equivalent fractions
L. Mult. • L. Num. 21x —————————————————— = ——— L.C.M 10 R. Mult. • R. Num. 12 • 2 —————————————————— = —————— L.C.M 10
Adding fractions that have a common denominator :
6.4 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:
21x + 12 • 2 21x + 24
———————————— = ————————
10 10
Equation at the end of step 6 :
(21x + 24) (18x - 3)
—————————— - ————————— ≥ 0
10 5
Step 7 :
Step 8 :
Pulling out like terms :
8.1 Pull out like factors :
21x + 24 = 3 • (7x + 8)
Step 9 :
Pulling out like terms :
9.1 Pull out like factors :
18x - 3 = 3 • (6x - 1)
Calculating the Least Common Multiple :
9.2 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 :
9.3 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 = 1
Right_M = L.C.M / R_Deno = 2
Making Equivalent Fractions :
9.4 Rewrite the two fractions into equivalent fractions
L. Mult. • L. Num. 3 • (7x+8) —————————————————— = —————————— L.C.M 10 R. Mult. • R. Num. 3 • (6x-1) • 2 —————————————————— = —————————————— L.C.M 10
Adding fractions that have a common denominator :
9.5 Adding up the two equivalent fractions
3 • (7x+8) - (3 • (6x-1) • 2) 30 - 15x
————————————————————————————— = ————————
10 10
Step 10 :
Pulling out like terms :
10.1 Pull out like factors :
30 - 15x = -15 • (x - 2)
Equation at the end of step 10 :
-15 • (x - 2)
————————————— ≥ 0
10
Step 11 :
11.1 Multiply both sides by 10
11.2 Divide both sides by -15
Remember to flip the inequality sign:
Solve Basic Inequality :
11.3 Add 2 to both sides
x ≤ 2
Inequality Plot :
11.4 Inequality plot for
-1.500 X + 3.000 ≤ 0
One solution was found :
x ≤ 2How did we do?
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