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 :
2-4/9*w-(w+7/6)<0
Step by step solution :
Step 1 :
7
Simplify —
6
Equation at the end of step 1 :
4 7
(2 - (— • w)) - (w + —) < 0
9 6
Step 2 :
Rewriting the whole as an Equivalent Fraction :
2.1 Adding a fraction to a whole
Rewrite the whole as a fraction using 6 as the denominator :
w w • 6
w = — = —————
1 6
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 :
2.2 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:
w • 6 + 7 6w + 7
————————— = ——————
6 6
Equation at the end of step 2 :
4 (6w + 7)
(2 - (— • w)) - ———————— < 0
9 6
Step 3 :
4
Simplify —
9
Equation at the end of step 3 :
4 (6w + 7)
(2 - (— • w)) - ———————— < 0
9 6
Step 4 :
Rewriting the whole as an Equivalent Fraction :
4.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using 9 as the denominator :
2 2 • 9
2 = — = —————
1 9
Adding fractions that have a common denominator :
4.2 Adding up the two equivalent fractions
2 • 9 - (4w) 18 - 4w
———————————— = ———————
9 9
Equation at the end of step 4 :
(18 - 4w) (6w + 7)
————————— - ———————— < 0
9 6
Step 5 :
Step 6 :
Pulling out like terms :
6.1 Pull out like factors :
18 - 4w = -2 • (2w - 9)
Calculating the Least Common Multiple :
6.2 Find the Least Common Multiple
The left denominator is : 9
The right denominator is : 6
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 3 | 2 | 1 | 2 |
| 2 | 0 | 1 | 1 |
| Product of all Prime Factors | 9 | 6 | 18 |
Least Common Multiple:
18
Calculating Multipliers :
6.3 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 = 2
Right_M = L.C.M / R_Deno = 3
Making Equivalent Fractions :
6.4 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. -2 • (2w-9) • 2 —————————————————— = ——————————————— L.C.M 18 R. Mult. • R. Num. (6w+7) • 3 —————————————————— = —————————— L.C.M 18
Adding fractions that have a common denominator :
6.5 Adding up the two equivalent fractions
-2 • (2w-9) • 2 - ((6w+7) • 3) 15 - 26w
—————————————————————————————— = ————————
18 18
Equation at the end of step 6 :
15 - 26w
———————— < 0
18
Step 7 :
7.1 Multiply both sides by 18
7.2 Multiply both sides by (-1)
Flip the inequality sign since you are multiplying by a negative number
26w-15 > 0
7.3 Divide both sides by 26
w-(15/26) > 0
Solve Basic Inequality :
7.4 Add 15/26 to both sides
w > 15/26
Inequality Plot :
7.5 Inequality plot for
-1.444 w + 0.833 < 0
One solution was found :
w > 15/26How did we do?
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