Find the values of $x$ that satisfy the following inequalities:

(i) $3 x^2-4<4 x$
(ii) $\frac{2 x-1}{x+2} \geq 1$
(iii) $2 x^2+8 x \leq 120$
(iv) $\left|\frac{x-4}{x}\right| \leq 3$
(v) $\frac{x}{x-2}>\frac{1}{x+1}$
(vi) $\left|\frac{x-2}{x+4}\right|>3$
(vii) $\left|\frac{2 x-1}{1-3 x}\right| \geq 0$
(viii) $\left|\frac{10 x-1}{4 x-6}\right|>-1

See the below works. (answers for q.vii & viii are at the attachment) ;

Answered by karmenchong | 2025-07-11

The solutions to the inequalities cover ranges of values for x based on various algebraic manipulations and sign analyses. Each inequality can be treated through critical points found by rearranging and testing intervals. Key methods used include factorization or rewriting of rational expressions, as well as consideration of absolute values and discontinuities.
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Answered by karmenchong | 2025-07-11