Linear Inequations — chapter notes

A linear inequation in one variable compares two linear expressions with one of the signs $<$, $\le$, $>$, $\ge$, for example $3x + 2 < 14$.

• $<$ and $>$ are strict: the boundary number is not a solution.
• $\le$ and $\ge$ include the boundary: $x = 3$ satisfies $x \le 3$ but not $x < 3$.

The replacement set (domain) is the set from which $x$ is taken: $\mathbb{N} = \{1, 2, 3, \ldots\}$, $\mathbb{W} = \{0, 1, 2, \ldots\}$, $\mathbb{Z} = \{\ldots, -1, 0, 1, \ldots\}$ or $\mathbb{R}$, or a set given in the question. The solution set is the subset whose members make the inequation true.

Example: $3x - 2 \le 7$ gives $x \le 3$: over $\mathbb{N}$, $\{1, 2, 3\}$; over $\mathbb{W}$, $\{0, 1, 2, 3\}$; over $\mathbb{Z}$, $\{\ldots, 1, 2, 3\}$.

An inequation keeps its solution set when you

• add or subtract the same number (or expression) on both sides;
• multiply or divide both sides by the same positive number.

Then solve as for an equation: remove brackets, clear fractions with the LCM (multiply every term), collect $x$ terms on one side and numbers on the other.

Example: $\dfrac{x}{3} - 1 \le \dfrac{x - 2}{4}$; multiply by $12$: $4x - 12 \le 3x - 6$, so $x \le 6$.

Multiplying or dividing both sides by a negative number reverses the sign: $<$ becomes $>$ and $\le$ becomes $\ge$.

Why: multiplying by $-1$ reflects the number line in $0$, so the order of any two numbers is turned round: $2 < 5$ but $-2 > -5$.

Example: $7 - 3x \ge -8 \Rightarrow -3x \ge -15 \Rightarrow x \le 5$ (dividing by $-3$).

$a < px + q \le b$ means $a < px + q$ and $px + q \le b$.

• When $x$ appears only in the middle, do the same step to all three parts; dividing by a negative number reverses both signs. Rewrite with the smaller number on the left.
• When $x$ appears in all three parts (e.g. $3x - 7 < 2x + 1 \le 4x + 6$), split it into two inequations and take the common part.

Example: $5 < 11 - 2x \le 17 \Rightarrow -6 < -2x \le 6 \Rightarrow 3 > x \ge -3$, i.e. $-3 \le x < 3$.

• $x \in \mathbb{N}$, $\mathbb{W}$ or $\mathbb{Z}$: separate filled dots at the members; three dots ($\ldots$) or an arrow shows that the set goes on.
• $x \in \mathbb{R}$: a thick line, with a filled circle at an included end point ($\le$, $\ge$), an open circle at an excluded end point ($<$, $>$), and an arrow if the set goes on for ever.
• Reading a graph: open circle $\to$ strict sign, filled circle $\to$ sign with "or equal to".

For $A = \{x : \ldots\}$ and $B = \{x : \ldots\}$, solve each separately.

• $A \cap B$: numbers in both, the overlap of the two graphs. Example: $x > -2$ and $x \le 4$ give $-2 < x \le 4$.
• $A \cup B$: numbers in at least one of them.
• If the graphs do not overlap ($x > 4$ and $x < 2$), $A \cap B = \phi$.