10.1 Linear Inequalities

Solve each of the following inequalities. Hence show on a number line

(a).

\(3x-2\leq -1\) \begin {align*} 3x-2 &\leq -1\\ \implies 3x & \leq -1+2\\ \implies x & \leq \frac {1}{3}\\\\ \therefore \quad SS: & \left (-\infty ,\frac {1}{3}\right ] \end {align*}

(b).

\(\frac {2}{3}x+5>3\) \begin {align*} \frac {2}{3}x &> 3-5\\ \implies \frac {2}{3}x & > -2\\ \implies x & > -3\\\\ \therefore \quad SS:& (-3,+\infty ) \end {align*}

(c).

\(2x+3\geq 5x-2\) \begin {align*} 2x+3 & \geq 5x-2\\ \implies 2x-5x & \geq -2-3\\ \implies -3x & \geq -5\\ \implies x & \leq \frac {-5}{-3} \quad \text {(dividing by $-3$ reverses $\geq $ to $\leq $)}\\ \implies x & \leq \frac {5}{3}\\\\ \therefore \quad SS:& \left (-\infty ,\frac {5}{3}\right ] \end {align*}

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