$(4x-5)/(3x+5) ≥ 3$
I multiply both sides by (3x+5), getting me:
$(4x-5) ≥ 3(3x+5)$
which simplifies to
$(4x-5) ≥ 9x + 15$
after solving for x, I get
$x ≤ -4$
But after testing through Wolframalpha, I am given:
$-4 ≤ x ≤ -5/3$
Which I don't really understand how they got. I tried multiplying the top side of the fraction by the denominator and then expanding the factored form, and then I rearranged everything once I had a quadratic equation, and I got $-5/3$ and $-2$ as answers, but still not what Wolframalpha got.
Would appreciate some help and insight into this.
Thanks.
$\endgroup$ 32 Answers
$\begingroup$If you multiply by a positive number the $\geq$ stays but when you multiply by a negative number sign changes to $\leq$.So you can take 2 cases when $3x+5\geq0$ and $3x+5<0$.Anyway it's better to go $$\frac{4x-5}{3x+5}-3\geq 0\\\frac{4x-5-9x-15}{3x+5}\geq0\\\frac{-5x-20}{3x+5}\geq0$$
$\endgroup$ $\begingroup$For $x>-\frac{5}{3}$:
$$4x-5 \geq 3(3x+5) \Rightarrow 4x-5 \geq 9x+15 \Rightarrow 5x \geq -20 \Rightarrow x \geq -4$$
For $x<-\frac{5}{3}$:
$$4x-5 \leq 3(3x+5) \Rightarrow 4x-5 \leq 9x+15 \Rightarrow \dots $$
$\endgroup$
