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 :
-1/6*x-(3/4+2/3*x)≤0
Step by step solution :
Step 1 :
2
Simplify —
3
Equation at the end of step 1 :
1 3 2
(0-(—•x))-(—+(—•x)) ≤ 0
6 4 3
Step 2 :
3
Simplify —
4
Equation at the end of step 2 :
1 3 2x
(0 - (— • x)) - (— + ——) ≤ 0
6 4 3
Step 3 :
Calculating the Least Common Multiple :
3.1 Find the Least Common Multiple
The left denominator is : 4
The right denominator is : 3
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 2 | 2 | 0 | 2 |
| 3 | 0 | 1 | 1 |
| Product of all Prime Factors | 4 | 3 | 12 |
Least Common Multiple:
12
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 = 3
Right_M = L.C.M / R_Deno = 4
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. 3 • 3 —————————————————— = ————— L.C.M 12 R. Mult. • R. Num. 2x • 4 —————————————————— = —————— L.C.M 12
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:
3 • 3 + 2x • 4 8x + 9
—————————————— = ——————
12 12
Equation at the end of step 3 :
1 (8x + 9)
(0 - (— • x)) - ———————— ≤ 0
6 12
Step 4 :
1
Simplify —
6
Equation at the end of step 4 :
1 (8x + 9)
(0 - (— • x)) - ———————— ≤ 0
6 12
Step 5 :
Calculating the Least Common Multiple :
5.1 Find the Least Common Multiple
The left denominator is : 6
The right denominator is : 12
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 2 | 1 | 2 | 2 |
| 3 | 1 | 1 | 1 |
| Product of all Prime Factors | 6 | 12 | 12 |
Least Common Multiple:
12
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 = 2
Right_M = L.C.M / R_Deno = 1
Making Equivalent Fractions :
5.3 Rewrite the two fractions into equivalent fractions
L. Mult. • L. Num. -x • 2 —————————————————— = —————— L.C.M 12 R. Mult. • R. Num. (8x+9) —————————————————— = —————— L.C.M 12
Adding fractions that have a common denominator :
5.4 Adding up the two equivalent fractions
-x • 2 - ((8x+9)) -10x - 9
————————————————— = ————————
12 12
Step 6 :
Pulling out like terms :
6.1 Pull out like factors :
-10x - 9 = -1 • (10x + 9)
Equation at the end of step 6 :
-10x - 9
———————— ≤ 0
12
Step 7 :
7.1 Multiply both sides by 12
7.2 Multiply both sides by (-1)
Flip the inequality sign since you are multiplying by a negative number
10x+9 ≥ 0
7.3 Divide both sides by 10
x+(9/10) ≥ 0
Solve Basic Inequality :
7.4 Subtract 9/10 from both sides
x ≥ -9/10
Inequality Plot :
7.5 Inequality plot for
-0.833 x - 0.750 ≤ 0
One solution was found :
x ≥ -9/10How did we do?
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