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 :
(x+3)/6-(x/4+1)>0
Step by step solution :
Step 1 :
x
Simplify —
4
Equation at the end of step 1 :
(x + 3) x
——————— - (— + 1) > 0
6 4
Step 2 :
Rewriting the whole as an Equivalent Fraction :
2.1 Adding a whole to a fraction
Rewrite the whole as a fraction using 4 as the denominator :
1 1 • 4
1 = — = —————
1 4
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:
x + 4 x + 4
————— = —————
4 4
Equation at the end of step 2 :
(x + 3) (x + 4)
——————— - ——————— > 0
6 4
Step 3 :
x + 3
Simplify —————
6
Equation at the end of step 3 :
(x + 3) (x + 4)
——————— - ——————— > 0
6 4
Step 4 :
Calculating the Least Common Multiple :
4.1 Find the Least Common Multiple
The left denominator is : 6
The right denominator is : 4
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 2 | 1 | 2 | 2 |
| 3 | 1 | 0 | 1 |
| Product of all Prime Factors | 6 | 4 | 12 |
Least Common Multiple:
12
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 = 2
Right_M = L.C.M / R_Deno = 3
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. (x+3) • 2 —————————————————— = ————————— L.C.M 12 R. Mult. • R. Num. (x+4) • 3 —————————————————— = ————————— L.C.M 12
Adding fractions that have a common denominator :
4.4 Adding up the two equivalent fractions
(x+3) • 2 - ((x+4) • 3) -x - 6
——————————————————————— = ——————
12 12
Step 5 :
Pulling out like terms :
5.1 Pull out like factors :
-x - 6 = -1 • (x + 6)
Equation at the end of step 5 :
-x - 6
—————— > 0
12
Step 6 :
6.1 Multiply both sides by 12
6.2 Multiply both sides by (-1)
Flip the inequality sign since you are multiplying by a negative number
x+6 < 0
Solve Basic Inequality :
6.3 Subtract 6 from both sides
x < -6
Inequality Plot :
6.4 Inequality plot for
-0.083 x - 0.500 > 0
One solution was found :
x < -6How did we do?
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