Question

Please na solve it please please.​

Answer 1

`\textsf{24.} \quad (1)/((x-2)(x-3))`

`\textsf{25.} \quad x=1, \quad x=-1`

Explanation:

Question 24

Given expression:

`(2)/((x-2)(x-3))+(2)/((x-1)(3-x))+(1)/((1-x)(2-x))`

When adding rationals with different denominators, make the denominators the same by finding the least common denominator.

Create a common denominator by:

`\textsf{Multiplying the first fraction by $(x-1)/(x-1)$}.`

`\textsf{Multiplying the second fraction by $(-(x-2))/(-(x-2))$}.`

`\textsf{Multiplying the third fraction by $(x-3)/(x-3)$}.`

`\implies (2)/((x-2)(x-3)) \cdot (x-1)/(x-1)+(2)/((x-1)(3-x))\cdot (-(x-2))/(-(x-2))+(1)/((1-x)(2-x))\cdot (x-3)/(x-3)`

`\implies (2(x-1))/((x-1)(x-2)(x-3))+(-2(x-2))/(-(x-1)(x-2)(3-x))+(x-3)/((1-x)(2-x)(x-3))`

`\implies (2(x-1))/((x-1)(x-2)(x-3))+(-2(x-2))/((x-1)(x-2)(-3-(-x)))+(x-3)/((-(-1+x))(-(-2+x))(x-3))`

`\implies (2(x-1))/((x-1)(x-2)(x-3))+(-2(x-2))/((x-1)(x-2)(x-3))+(x-3)/((x-1)(x-2)(x-3))`

`\textsf{Apply the fraction rule} \quad (a)/(d)+(b)/(d)+(c)/(d)=(a+b+c)/(d):`

`\implies (2(x-1)-2(x-2)+(x-3))/((x-1)(x-2)(x-3))`

Expand the numerator:

`\implies (2x-2-2x+4+x-3)/((x-1)(x-2)(x-3))`

`\implies (x-1)/((x-1)(x-2)(x-3))`

Cancel the common factor (x - 1):

`\implies (1)/((x-2)(x-3))`

Question 25

Given equation:

`(4x-1)(x+2)=2+7x`

Expand the left side:

`\implies 4x^2+7x-2=2+7x`

Subtract 7x from both sides:

`\implies 4x^2+7x-2-7x=2+7x-7x`

`\implies 4x^2-2=2`

Add 2 to both sides:

`\implies 4x^2-2+2=2+2`

`\implies 4x^2=4`

Divide both sides by 4:

`\implies (4x^2)/(4)=(4)/(4)`

`\implies x^2=1`

Square root both sides:

`\implies √(x^2)=√(1)`

`\implies x= \pm 1`

Therefore, the solutions are:

`x=1, \quad x=-1`

Answer 2

Solution given to both questions below, they reveal the given answers.

The steps are self-explanatory, let me know if anything is unclear.

Question 24

• 
  `\cfrac{2}{(x-2)(x-3)} +\cfrac{2}{(x-1)(3-x)}+\cfrac{1}{(1-x)(2-x)} =`

• 
  `\cfrac{2}{(x-2)(x-3)} -\cfrac{2}{(x-1)(x-3)}+\cfrac{1}{(x-1)(x-2)} =`

• 
  `\cfrac{2(x-1)}{(x-1)(x-2)(x-3)} -\cfrac{2(x-2)}{(x-1)(x-2)(x-3)}+\cfrac{x-3}{(x-1)(x-2)(x-3)} =`

• 
  `\cfrac{2x-2-2x+4+x-3}{(x-1)(x-2)(x-3)} =`

• 
  `\cfrac{x-1}{(x-1)(x-2)(x-3)} =`

• 
  `\cfrac{1}{(x-2)(x-3)}`

Question 25

• 
  `(4x - 1)(x + 2) = 2 + 7x`
• 
  `4x^2+8x-x-2=2+7x`
• 
  `4x^2+7x-2=2+7x`
• 
  `4x^2=4`
• 
  `x^2=1`
• 
  `x=√(1)`
• 
  `x=\pm1`