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 equal sign from both sides of the inequality :
b+7/8-(1/6)≥0
Step by step solution :
Step 1 :
1
Simplify —
6
Equation at the end of step 1 :
7 1
(b + —) - — ≥ 0
8 6
Step 2 :
7
Simplify —
8
Equation at the end of step 2 :
7 1
(b + —) - — ≥ 0
8 6
Step 3 :
Rewriting the whole as an Equivalent Fraction :
3.1 Adding a fraction to a whole
Rewrite the whole as a fraction using 8 as the denominator :
b b • 8
b = — = —————
1 8
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 :
3.2 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:
b • 8 + 7 8b + 7
————————— = ——————
8 8
Equation at the end of step 3 :
(8b + 7) 1
———————— - — ≥ 0
8 6
Step 4 :
Calculating the Least Common Multiple :
4.1 Find the Least Common Multiple
The left denominator is : 8
The right denominator is : 6
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 2 | 3 | 1 | 3 |
| 3 | 0 | 1 | 1 |
| Product of all Prime Factors | 8 | 6 | 24 |
Least Common Multiple:
24
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 = 3
Right_M = L.C.M / R_Deno = 4
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. (8b+7) • 3 —————————————————— = —————————— L.C.M 24 R. Mult. • R. Num. 4 —————————————————— = —— L.C.M 24
Adding fractions that have a common denominator :
4.4 Adding up the two equivalent fractions
(8b+7) • 3 - (4) 24b + 17
———————————————— = ————————
24 24
Equation at the end of step 4 :
24b + 17
———————— ≥ 0
24
Step 5 :
5.1 Multiply both sides by 24
5.2 Divide both sides by 24
b+(17/24) ≥ 0
Solve Basic Inequality :
5.3 Subtract 17/24 from both sides
b ≥ -17/24
Inequality Plot :
5.4 Inequality plot for
b + 0.708 ≥ 0
One solution was found :
b ≥ -17/24How did we do?
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