Question

Please could you explain this step by step I’m really stuck.

Answer 1

x = -2.5
x = 2

Explanation:

Hello!

We can remove the denominators by multiplying everything by it.

Solve:

• 
  `(5)/(x + 3) + \frac4{x + 2} = 2`
• 
  `5 + (4(x + 3))/(x + 2) = 2(x + 3)`
• 
  `5 + (4x + 12)/(x + 2) = 2x + 6`
• 
  `5(x + 2) +4x + 12 = (2x + 6)(x + 2)`
• 
  `5x + 10 + 4x + 12 = 2x^2 + 6x + 4x + 12`
• 
  `9x + 22 = 2x^2 + 10x + 12`

Let's make one side equal to 0.

• 
  `0 = 2x^2 + x - 10`

Solve by factoring, and then the zero product property.

• 
  `0 = 2x^2 + x - 10\\`

Multiply 2 and -10. You get -20. Think of two number that multiply to -20, and add to 1. If we think about the standard form of a quadratic,
`ax^2 + bx + c`, think two numbers that equal "ac" and add up to "b".

• 
  `0 = 2x^2 - 4x + 5x - 10`
• 
  `0 = 2x(x - 2) + 5(x - 2)`
• 
  `0 = (2x + 5)(x - 2)`

Using the zero product property, set each factor to 0 and solve for x.

1. 2x + 5 = 0
   2x = -5
   x = -5/2 = -2.5
2. x - 2 = 0
   x = 2

The two answers are x = -2.5, and x = 2.

Answer 2

x = -5/2, 2

Explanation:

Given:

`\displaystyle \large{(5)/(x+3)+(4)/(x+2) = 2}`

With restriction that x-values cannot be -3 and -2 else it will turn the expression as in undefined.

First, multiply both sides by (x+3)(x+2) to get rid of the denominator.

`\displaystyle \large{(5)/(x+3)\cdot (x+2)(x+3)+(4)/(x+2) \cdot (x+2)(x+3) = 2 \cdot (x+2)(x+3)}\\\\\displaystyle \large{5(x+2)+4(x+3) = 2(x+2)(x+3)}`

Simplify/Expand in:

`\displaystyle \large{5x+10+4x+12 = 2(x^2+5x+6)}\\\\\displaystyle \large{9x+22=2x^2+10x+12}`

Arrange the terms or expression in quadratic equation:

`\displaystyle \large{0=2x^2+10x+12-9x-22}\\\\\displaystyle \large{2x^2+x-10=0}`

Factor the expression:

`\displaystyle \large{(2x+5)(x-2)=0}`

Solve like linear equation as we get:

`\displaystyle \large{x=-(5)/(2), 2}`

Since both x-values are not exact -2 or -3 - therefore, these values are valid. Hence, x = -5/2, 2