Algebra - Inequalities

Inequality Symbols and Real-World Constraints

An inequality uses symbols to show how quantities compare:

• $<$ means strictly less than; $>$ means strictly greater than.
• $\leq$ means less than or equal to; $\geq$ means greater than or equal to.
• For $<$ and $>$, the pointed end always faces the smaller value.
• Strict inequalities ($<$ and $>$) exclude the boundary; $\leq$ and $\geq$ include it.

Translating Words into Inequalities

• "At least" and "a minimum of" mean $\geq$; "at most" and "no more than" mean $\leq$.
• "More than" means $>$; "less than" and "fewer than" mean $<$.
• Physical counts cannot be negative, so an expression like $184 - n \geq 0$ gives $n \leq 184$.
• If €5 plus €3 per visit is capped at €20, then $5 + 3n \leq 20$ allows at most 5 visits.

Solving Linear and Double Inequalities

You solve inequalities using standard equation balance steps, with one golden rule: whenever you multiply or divide both sides by a negative number, reverse the inequality symbol.

• Gather variable terms on one side first: $5x - 3 \geq 2x + 9 \implies 3x \geq 12 \implies x \geq 4$.
• For $3 - 4x < 11$, dividing $-4x < 8$ by $-4$ reverses the symbol to give $x > -2$.
• Clear algebraic fractions early by multiplying every term across by the lowest common denominator.

Double Inequalities

A double inequality bounds an expression between two values. Apply every operation to all three parts simultaneously to isolate the variable.

Graphing on the Number Line

The number set specified for the variable dictates how you plot the solution set on the number line.

For real numbers ($x \in \mathbb{R}$):

• Use an open circle ($\circ$) at an excluded boundary ($<$ or $>$).
• Use a closed dot ($\bullet$) at an included boundary ($\leq$ or $\geq$).
• For $x > -2$, place an open circle at $-2$ and shade right with an arrow.
• For $-1 \leq x < 5$, shade between a closed dot at $-1$ and open circle at $5$.

Inequalities from Graphs: f(x) < g(x)

At Higher Level, you must interpret approximate solution sets of inequalities directly from given function plots.

• Intersection points show where $f(x) = g(x)$.
• The inequality $f(x) < g(x)$ holds where the graph of $f(x)$ lies strictly below $g(x)$.
• Check every domain region to see where $f(x)$ sits below $g(x)$.
• State solution sets as $x$-intervals, such as $-1 < x < 2$.
• If solutions form disconnected branches, write them as separate ranges, such as $x < -1$ or $x > 2$.

Key terms

Inequality

A statement comparing two quantities using $<$, $>$, $\leq$, or $\geq$.

Solution Set

The collection of all values that make an inequality true when substituted for the variable.

Double Inequality

A three-part mathematical statement where an algebraic expression is bounded between two boundary values.

Open Circle

An unfilled circle plotted on a number line to show that a boundary value is excluded from the solution set.

Closed Dot

A solid, filled dot plotted on a number line to indicate that a boundary value is included in the solution set.