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 :
1/2*x-(1/3*(2*x-6)+3)<0
Step by step solution :
Step 1 :
1
Simplify —
3
Equation at the end of step 1 :
1 1
(— • x) - ((— • (2x - 6)) + 3) < 0
2 3
Step 2 :
Step 3 :
Pulling out like terms :
3.1 Pull out like factors :
2x - 6 = 2 • (x - 3)
Equation at the end of step 3 :
1 2 • (x - 3)
(— • x) - (——————————— + 3) < 0
2 3
Step 4 :
Rewriting the whole as an Equivalent Fraction :
4.1 Adding a whole to a fraction
Rewrite the whole as a fraction using 3 as the denominator :
3 3 • 3
3 = — = —————
1 3
Equivalent fraction : The fraction thus generated looks different but has the same value as the whole
Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator
Adding fractions that have a common denominator :
4.2 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:
2 • (x-3) + 3 • 3 2x + 3
————————————————— = ——————
3 3
Equation at the end of step 4 :
1 (2x + 3)
(— • x) - ———————— < 0
2 3
Step 5 :
1
Simplify —
2
Equation at the end of step 5 :
1 (2x + 3)
(— • x) - ———————— < 0
2 3
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
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. x • 3 —————————————————— = ————— L.C.M 6 R. Mult. • R. Num. (2x+3) • 2 —————————————————— = —————————— L.C.M 6
Adding fractions that have a common denominator :
6.4 Adding up the two equivalent fractions
x • 3 - ((2x+3) • 2) -x - 6
———————————————————— = ——————
6 6
Step 7 :
Pulling out like terms :
7.1 Pull out like factors :
-x - 6 = -1 • (x + 6)
Equation at the end of step 7 :
-x - 6
—————— < 0
6
Step 8 :
8.1 Multiply both sides by 6
8.2 Multiply both sides by (-1)
Flip the inequality sign since you are multiplying by a negative number
x+6 > 0
Solve Basic Inequality :
8.3 Subtract 6 from both sides
x > -6
Inequality Plot :
8.4 Inequality plot for
-0.167 x - 1.000 < 0
One solution was found :
x > -6How did we do?
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