## solve the equation 3/4 X+-2X=-1/4+1/2X+5​

Question

solve the equation 3/4 X+-2X=-1/4+1/2X+5​

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40 mins 2021-09-15T22:56:00+00:00 1 Answer 0

x = -19/7 = -2.714

Step-by-step explanation:

Step  1  :

1

Simplify   —

2

Equation at the end of step  1  :

3             1   1

((—•x)-2x)-(((0-—)+(—•x))+5)  = 0

4             4   2

Step  2  :

1

Simplify   —

4

Equation at the end of step  2  :

3             1  x

((—•x)-2x)-(((0-—)+—)+5)  = 0

4             4  2

Step  3  :

Calculating the Least Common Multiple :

3.1    Find the Least Common Multiple

The left denominator is :       4

The right denominator is :       2

Number of times each prime factor

appears in the factorization of:

Prime

Factor   Left

Denominator   Right

Denominator   L.C.M = Max

{Left,Right}

2 2 1 2

Product of all

Prime Factors  4 2 4

Least Common Multiple:

4

Calculating Multipliers :

3.2    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 = 2

Making Equivalent Fractions :

3.3      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.      -1

——————————————————  =   ——

L.C.M             4

R. Mult. • R. Num.      x • 2

——————————————————  =   —————

L.C.M               4

Adding fractions that have a common denominator :

3.4       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:

-1 + x • 2     2x – 1

——————————  =  ——————

4            4

Equation at the end of step  3  :

3                 (2x – 1)

((— • x) –  2x) –  (———————— +  5)  = 0

4                    4

Step  4  :

Rewriting the whole as an Equivalent Fraction :

4.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  4  as the denominator :

5     5 • 4

5 =  —  =  —————

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 :

4.2       Adding up the two equivalent fractions

(2x-1) + 5 • 4     2x + 19

——————————————  =  ———————

4               4

Equation at the end of step  4  :

3                (2x + 19)

((— • x) –  2x) –  —————————  = 0

4                    4

Step  5  :

3

Simplify   —

4

Equation at the end of step  5  :

3                (2x + 19)

((— • x) –  2x) –  —————————  = 0

4                    4

Step  6  :

Rewriting the whole as an Equivalent Fraction :

6.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  4  as the denominator :

2x     2x • 4

2x =  ——  =  ——————

1        4

Adding fractions that have a common denominator :

6.2       Adding up the two equivalent fractions

3x – (2x • 4)     -5x

—————————————  =  ———

4            4

Equation at the end of step  6  :

-5x    (2x + 19)

——— –  —————————  = 0

4         4

Step  7  :

Adding fractions which have a common denominator :

7.1       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:

-5x – ((2x+19))     -7x – 19

———————————————  =  ————————

4               4

Step  8  :

Pulling out like terms :

8.1     Pull out like factors :

-7x – 19  =   -1 • (7x + 19)

Equation at the end of step  8  :

-7x – 19

————————  = 0

4

Step  9  :

When a fraction equals zero :

9.1    When a fraction equals zero …

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here’s how:

-7x-19

—————— • 4 = 0 • 4

4

Now, on the left hand side, the  4  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :

-7x-19  = 0

Solving a Single Variable Equation :

9.2      Solve  :    -7x-19 = 0

Add  19  to both sides of the equation :

-7x = 19

Multiply both sides of the equation by (-1) :  7x = -19

Divide both sides of the equation by 7:

x = -19/7 = -2.714

One solution was found :

x = -19/7 = -2.714

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