Question

Solving equations, thank you!

Answer 1

x = -4 or x = 1

Explanation:

`(x)/(2x-3)-(8-3x)/(4x^2-9)=0\\\\\bold{DOMAIN:}\\\\2x-3\\eq0\ \wedge\ 4x^2-9\\eq0\\\\2x-3\\eq0\qquad\text{add 3 to both sides}\\2x\\eq3\qquad\text{divide both sides by 2}\\\boxed{x\\eq1.5}\\\\4x^2-9\\eq0\qquad\text{add 9 to both sides}\\4x^2\\eq9\qquad\text{divide both sides by 4}\\x^2\\eq2.25\\x\\eq\pm√(2.25)\\\boxed{x\\eq-1.5\ \wedge\ x\\eq1.5}`

`(x)/(2x-3)-(8-3x)/(4x^2-9)=0\\\\(x)/(2x-3)-(8-3x)/((2x)^2-3^2)=0\qquad\text{use}\ a^2-b^2=(a-b)(a+b)\\\\(x)/(2x-3)-(8-3x)/((2x-3)(2x+3))=0\\\\(x(2x+3))/((2x-3)(2x+3))-(8-3x)/((2x-3)(2x+3))=0\\\\(x(2x+3)-(8-3x))/(4x^2-9)=0`

`\text{The fraction is equal to 0 if the numerator is equal to 0. Therefore}\\\\(x(2x+3)-(8-3x))/(4x^2-9)=0\iff x(2x-3)-(8-3x)=0\\\\\text{use the distributive property}\\\\(x)(2x)+(x)(3)-8-(-3x)=0\\\\2x^2+3x-8+3x=0\qquad\text{combine like terms}\\\\2x^2+(3x+3x)-8=0\\\\2x^2+6x-8=0\qquad\text{divide both sides by 2}\\\\x^2+3x-4=0\\\\x^2+4x-1x-4=0\\\\x(x+4)-1(x+4)=0\\\\(x+4)(x-1)=0\iff x+4=0\ \vee\ x-1=0\\\\x+4=0\qquad\text{subtract 4 from both sides}\\x=-4\in D\\\\x-1=0\qquad\text{add 1 to both sides}\\x=1`