Double Inequalities (Leaving Cert Mathematics): Revision Notes

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Double Inequalities

In some cases we may encounter inequalities with two inequality signs. In such cases, you can solve both inequalities separately.

Example

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Solve the inequality

3x≤5x+3≤233x\le 5x+3 \le23

Separate the inequalities :

3x≤5x+3⏟≤23\underbrace{3x\le 5x+3}_{} \le23

Solve :

3x≤5x+3−2x≤3(−5x)x≥−32(÷(−2), flip sign) \begin{align*} 3x &\le 5x+3 & \\\\ -2x &\le 3 & \text{\footnotesize\textcolor{gray}{(\(-5x\))}} \\\\ x &\ge -\tfrac{3}{2} &\text{\footnotesize\textcolor{gray}{(\( \div (-2) \), flip sign)}} \\ \end{align*}

Separate the second inequality :

3x≤5x+3≤23⏟3x\le \underbrace{5x+3 \le23}

Solve :

5x+3≤235x≤20(−3)x≤4(÷5) \begin{align*} 5x+3 &\le23 & \\\\ 5x &\le20 & \text{\footnotesize\textcolor{gray}{(\(-3\))}} \\\\ x &\le 4 &\text{\footnotesize\textcolor{gray}{(\( \div 5 \))}} \\ \end{align*}

We have solved for two regions for which the inequality will be satisfied, xx needs to be greater (or equal) to −32-\tfrac{3}{2} but less than (or equal) to 44. Combine the two regions :

−32≤x≤4-\tfrac{3}{2} \le x \le 4

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Inequalities

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Inequality Proofs & Discriminants

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Inequalities

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Inequality Proofs & Discriminants

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Inequalities

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