We have an answer to "Solve rational equations and inequalitiesa) xx+3=8x+6b) 22x−3=3xx+11c) 7x+3=x4d)..."

blestimd4pz

Answered question

2022-03-22

Solve rational equations and inequalities
a)

\frac{x}{x + 3} = \frac{8}{x + 6}

\frac{2}{2 x - 3} = \frac{3 x}{x + 11}

\frac{7}{x + 3} = \frac{x}{4}

\frac{x - 3}{x + 2} \geq 0

Ishaan Stout

Beginner2022-03-23Added 14 answers

\frac{x}{x + 3} = \frac{8}{x + 6}

x (x + 6) = 8 (x + 3)

x^{2} + 6 x = 8 x + 24

x^{2} + 6 x - 8 x - 24 = 0

x^{2} - 2 x - 24 = 0

x^{2} - 6 x + 4 x - 24 = 0

x (x - 6) + 4 (x - 6) = 0

(x - 6) (x + 4) = 0

∴ x - 6 = 0

x + 4 = 0

x = 6

x = - 4

\frac{2}{2 x - 3} = \frac{3 x}{x + 11}

2 x + 22 = 3 x (2 x - 3)

2 x + 22 = 6 x^{2} - 9 x

6 x^{2} - 9 x - 2 x - 22 = 0

6 x^{2} - 11 x - 22 = 0

Here,

a = 6

b = - 11

c = - 22

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

x = \frac{- (- 11) \pm \sqrt{{(- 11)}^{2} - 4 \times 6 \times (- 22)}}{2 \times 6}

= \frac{11 \pm \sqrt{121 + 528}}{12}

= \frac{11 \pm \sqrt{649}}{12}

Thus,

x = \frac{11 - \sqrt{649}}{12}

\frac{11 + \sqrt{649}}{12}

Roy Brady

Beginner2022-03-24Added 19 answers

\frac{7}{x + 3} = \frac{x}{4}

21 = x^{2} + 3 x

x^{2} + 3 x - 21 = 0

a = 1

b = 3

c = - 21

x = \frac{- 3 \pm \sqrt{3^{2} - 4 \times 1 \times (- 21)}}{2 \times 1}

= \frac{- 3 \pm \sqrt{9 + 84}}{2}

= \frac{- 3 \pm \sqrt{93}}{2}

Thus,

x = \frac{- 3 + \sqrt{93}}{2}

x == \frac{- 3 - \sqrt{93}}{2}

\frac{x - 3}{x + 2} \geq 0

x - 3 \geq 0

x - 3 + 3 \geq 0 + 3

x \geq 3

Hence,

x \in [3, \infty)

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