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Algebra

# Two-sided Linear Inequalities

What is the greatest integer value of $$x$$ that satisfies the inequality $$-5 \leq 2x - 3 < 3?$$

Solve the simultaneous inequalities

\begin{align} -3x + 4 &\ge 1 \\ 2x - 5 &< 5x + 1. \end{align}

Solve the inequalities $$\dfrac{x-2}{3} < \dfrac{x+5}{4} < \dfrac{x-1}{2}.$$

The minimum value of $$x$$ that satisfies the inequality $$\dfrac{x - 5}{11 x - 4} \geq 3$$ can be expressed as $$\dfrac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$\dfrac{a}{b}?$$

If the range of $$x$$ represented by the arrows in the number line above is equivalent to $$x - a < -x - 1 \le x + b,$$ what is $$a + b ?$$

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