Solution - Adding, subtracting and finding the least common multiple
Step by Step Solution
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "2.3" was replaced by "(23/10)". 3 more similar replacement(s)
Rearrange:
Rearrange the equation by subtracting what is to the right of the less equal sign from both sides of the inequality :
-(12/10)*x-(65/10)*x-((23/10)*x+5)≤0
Step by step solution :
Step 1 :
23
Simplify ——
10
Equation at the end of step 1 :
12 65 23
((0-(——•x))-(——•x))-((——•x)+5) ≤ 0
10 10 10
Step 2 :
Rewriting the whole as an Equivalent Fraction :
2.1 Adding a whole to a fraction
Rewrite the whole as a fraction using 10 as the denominator :
5 5 • 10
5 = — = ——————
1 10
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:
23x + 5 • 10 23x + 50
———————————— = ————————
10 10
Equation at the end of step 2 :
12 65 (23x+50)
((0-(——•x))-(——•x))-———————— ≤ 0
10 10 10
Step 3 :
13
Simplify ——
2
Equation at the end of step 3 :
12 13 (23x+50)
((0-(——•x))-(——•x))-———————— ≤ 0
10 2 10
Step 4 :
6
Simplify —
5
Equation at the end of step 4 :
6 13x (23x + 50)
((0 - (— • x)) - ———) - —————————— ≤ 0
5 2 10
Step 5 :
Calculating the Least Common Multiple :
5.1 Find the Least Common Multiple
The left denominator is : 5
The right denominator is : 2
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 5 | 1 | 0 | 1 |
| 2 | 0 | 1 | 1 |
| Product of all Prime Factors | 5 | 2 | 10 |
Least Common Multiple:
10
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 = 2
Right_M = L.C.M / R_Deno = 5
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. -6x • 2 —————————————————— = ——————— L.C.M 10 R. Mult. • R. Num. 13x • 5 —————————————————— = ——————— L.C.M 10
Adding fractions that have a common denominator :
5.4 Adding up the two equivalent fractions
-6x • 2 - (13x • 5) -77x
——————————————————— = ————
10 10
Equation at the end of step 5 :
-77x (23x + 50)
———— - —————————— ≤ 0
10 10
Step 6 :
Adding fractions which have a common denominator :
6.1 Adding fractions which have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
-77x - ((23x+50)) -100x - 50
————————————————— = ——————————
10 10
Step 7 :
Pulling out like terms :
7.1 Pull out like factors :
-100x - 50 = -50 • (2x + 1)
Equation at the end of step 7 :
-50 • (2x + 1)
—————————————— ≤ 0
10
Step 8 :
8.1 Multiply both sides by 10
8.2 Divide both sides by -50
Remember to flip the inequality sign:
8.3 Divide both sides by 2
x+(1/2) ≥ 0
Solve Basic Inequality :
8.4 Subtract 1/2 from both sides
x ≥ -1/2
Inequality Plot :
8.5 Inequality plot for
-10.000 X - 5.000 ≥ 0
One solution was found :
x ≥ -1/2How did we do?
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