---
title: "Equations and inequalities: cheat sheet"
subtitle: "Algebra Foundations"
---

# Equations and inequalities

> The words and methods from this unit, with an example of each.

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**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.

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

_The two sides as lines meet at x = 8_

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**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.

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

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**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.

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

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**inequality on a number line**: Open circle for $<$ and $>$ (the end is not included), closed for $\le$ and $\ge$; shade towards the solutions.

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

_x < 5: an open circle at 5, shaded left_

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