Equations and inequalities: cheat sheet

Algebra Foundations · The words and methods from the lessons, with an example of each.

x on both sides

Expand any brackets. Take the smaller $x$ term from both sides, undo the numbers with the inverse (add to undo $-$ ), then divide.

3x + 7 = 5x - 9 \;\Rightarrow\; 16 = 2x \;\Rightarrow\; x = 8

check by substituting

Put the answer into both sides of the first equation. Equal sides: right. Unequal: look for the first step that went wrong.

3(8) + 7 = 31 = 5(8) - 9

solving an inequality

Do the same to both sides, as for an equation. Multiplying or dividing by a negative **reverses** the sign ( $2 < 5$ but $-2 > -5$ ). Check a value from your answer.

-2x > 6 \;\Rightarrow\; x < -3

inequality on a number line

Open circle for $<$ and $>$ (the end is not included), closed for $\le$ and $\ge$ ; shade towards the solutions.

x < 5:\ \circ \text{ at } 5,\ \text{shaded left}