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/3-1/4-(x/4+1)<0
Step by step solution :
Step 1 :
x
Simplify —
4
Equation at the end of step 1 :
x 1 x
(— - —) - (— + 1) < 0
3 4 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 1 (x + 4)
(— - —) - ——————— < 0
3 4 4
Step 3 :
1
Simplify —
4
Equation at the end of step 3 :
x 1 (x + 4)
(— - —) - ——————— < 0
3 4 4
Step 4 :
x
Simplify —
3
Equation at the end of step 4 :
x 1 (x + 4)
(— - —) - ——————— < 0
3 4 4
Step 5 :
Calculating the Least Common Multiple :
5.1 Find the Least Common Multiple
The left denominator is : 3
The right denominator is : 4
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 3 | 1 | 0 | 1 |
| 2 | 0 | 2 | 2 |
| Product of all Prime Factors | 3 | 4 | 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 = 4
Right_M = L.C.M / R_Deno = 3
Making Equivalent Fractions :
5.3 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 • 4 —————————————————— = ————— L.C.M 12 R. Mult. • R. Num. 3 —————————————————— = —— L.C.M 12
Adding fractions that have a common denominator :
5.4 Adding up the two equivalent fractions
x • 4 - (3) 4x - 3
——————————— = ——————
12 12
Equation at the end of step 5 :
(4x - 3) (x + 4)
———————— - ——————— < 0
12 4
Step 6 :
Calculating the Least Common Multiple :
6.1 Find the Least Common Multiple
The left denominator is : 12
The right denominator is : 4
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 2 | 2 | 2 | 2 |
| 3 | 1 | 0 | 1 |
| Product of all Prime Factors | 12 | 4 | 12 |
Least Common Multiple:
12
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 = 1
Right_M = L.C.M / R_Deno = 3
Making Equivalent Fractions :
6.3 Rewrite the two fractions into equivalent fractions
L. Mult. • L. Num. (4x-3) —————————————————— = —————— L.C.M 12 R. Mult. • R. Num. (x+4) • 3 —————————————————— = ————————— L.C.M 12
Adding fractions that have a common denominator :
6.4 Adding up the two equivalent fractions
(4x-3) - ((x+4) • 3) x - 15
———————————————————— = ——————
12 12
Equation at the end of step 6 :
x - 15
—————— < 0
12
Step 7 :
7.1 Multiply both sides by 12
Solve Basic Inequality :
7.2 Add 15 to both sides
x < 15
Inequality Plot :
7.3 Inequality plot for
0.083 x - 1.250 < 0
One solution was found :
x < 15How did we do?
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