Equations & Inequalities

Section 1 of 2

Solving Equations

An equation is a question and the answer is a number. Solving a linear equation means pinning $x$ down to a single value.

The move

When you move a term across the equals, its sign flips: $+ \leftrightarrow -$ and $\times \leftrightarrow \div$. The equals is the pivot — cross it and the operation reverses.

Worked example — one step

Solve $10x = 70$ and $x + 5 = 22$.

$10x = 70 \Rightarrow x = \tfrac{70}{10} = 7$

$x + 5 = 22 \Rightarrow x = 22 - 5 = 17$

$x = 7$ and $x = 17$

Worked example — two steps

Solve $2x - 1 = 9$.

$2x = 9 + 1 = 10$

$x = 5$

Worked example — x on both sides

Solve $5x - 3 = 3x + 9$.

$5x - 3x = 9 + 3$

$2x = 12$

$x = 6$

You try

Solve $5(x - 1) = 3(x + 7)$.

Multiply out both brackets first, then gather the $x$’s on one side.

$5x - 5 = 3x + 21$

$5x - 3x = 21 + 5$

$2x = 26$

$x = 13$

Section 2 of 2

Solving Inequalities

An inequality says one side is greater than, less than, or at most the other ($<, >, \le, \ge$). Solving one is almost identical to an equation — with one extra rule.

Change–change

Same recipe as equations. But whenever you multiply or divide by a negative to free $x$, the inequality flips: change the sign of the coefficient and change the direction. ($3 > 2$, but $-3 < -2$.)

Worked example — straightforward

Solve $3x - 1 \le 14$ and $5x - 3 \ge 27$.

$3x \le 15 \Rightarrow x \le 5$

$5x \ge 30 \Rightarrow x \ge 6$

$x \le 5$ and $x \ge 6$

Worked example — a negative coefficient

Solve $1 - 2x \ge 17$.

$-2x \ge 17 - 1 = 16$

$2x \le -16$ (change–change)

$x \le -8$

You try

Solve $2(x - 1) \le 4(x + 3)$.

Multiply out, gather $x$’s. If the $x$ coefficient goes negative, apply change–change.

$2x - 2 \le 4x + 12$

$2x - 4x \le 12 + 2$

$-2x \le 14 \Rightarrow 2x \ge -14$ (change–change)

$x \ge -7$

That’s Class 6.

Solving equations by flipping signs across the equals, and inequalities by the same recipe — plus change–change for a negative. Class 7: simultaneous equations.