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 :
2*x-1/2-(7*x+7/6)≥0
Step by step solution :
Step 1 :
7
Simplify —
6
Equation at the end of step 1 :
1 7
(2x - —) - (7x + —) ≥ 0
2 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 :
7x 7x • 6
7x = —— = ——————
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:
7x • 6 + 7 42x + 7
—————————— = ———————
6 6
Equation at the end of step 2 :
1 (42x + 7)
(2x - —) - ————————— ≥ 0
2 6
Step 3 :
1
Simplify —
2
Equation at the end of step 3 :
1 (42x + 7)
(2x - —) - ————————— ≥ 0
2 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 2 as the denominator :
2x 2x • 2
2x = —— = ——————
1 2
Adding fractions that have a common denominator :
4.2 Adding up the two equivalent fractions
2x • 2 - (1) 4x - 1
———————————— = ——————
2 2
Equation at the end of step 4 :
(4x - 1) (42x + 7)
———————— - ————————— ≥ 0
2 6
Step 5 :
Step 6 :
Pulling out like terms :
6.1 Pull out like factors :
42x + 7 = 7 • (6x + 1)
Calculating the Least Common Multiple :
6.2 Find the Least Common Multiple
The left denominator is : 2
The right denominator is : 6
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 2 | 1 | 1 | 1 |
| 3 | 0 | 1 | 1 |
| Product of all Prime Factors | 2 | 6 | 6 |
Least Common Multiple:
6
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 = 3
Right_M = L.C.M / R_Deno = 1
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. (4x-1) • 3 —————————————————— = —————————— L.C.M 6 R. Mult. • R. Num. 7 • (6x+1) —————————————————— = —————————— L.C.M 6
Adding fractions that have a common denominator :
6.5 Adding up the two equivalent fractions
(4x-1) • 3 - (7 • (6x+1)) -30x - 10
————————————————————————— = —————————
6 6
Step 7 :
Pulling out like terms :
7.1 Pull out like factors :
-30x - 10 = -10 • (3x + 1)
Equation at the end of step 7 :
-10 • (3x + 1)
—————————————— ≥ 0
6
Step 8 :
8.1 Multiply both sides by 6
8.2 Divide both sides by -10
Remember to flip the inequality sign:
8.3 Divide both sides by 3
x+(1/3) ≤ 0
Solve Basic Inequality :
8.4 Subtract 1/3 from both sides
x ≤ -1/3
Inequality Plot :
8.5 Inequality plot for
-5.000 X - 1.667 ≤ 0
One solution was found :
x ≤ -1/3How did we do?
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