IP EMaths Notes (Upper Sec, Year 3-4): 04) Linear Equations and Inequalities

Linear manipulations appear everywhere - simultaneous systems, variation models, and coordinate geometry. Practice consistent isolation and be fluent with inequality notation.

Key reminders

• Keep the equation balanced: whatever operation you apply to one side must be applied to the other.
• When multiplying or dividing an inequality by a negative number, reverse the inequality symbol.
• Present solution sets with interval notation or on a number line where appropriate.

Worked example - Solve and interpret

Solve the system \[ \begin{aligned} 2x - 3 &= 7, \ 5 - x &> 2. \end{aligned} \]

1. Equation: add 3 to both sides: \(2x = 10 \Rightarrow x = 5\).
2. Inequality: subtract 5 from both sides to get \(-x > -3\).
3. Multiply by \(-1\) (and flip the sign): \(x < 3\).
4. Combine: the equation demands \(x = 5\) but the inequality demands \(x < 3\). There is no common solution, so the system is inconsistent.

Try this

Solve \(7 - 2(3 - y) \leq 4y + 1\) and express the solution set for \(y\) on a number line.