Linear inequalities Knowledge-Based Study

Study notes

1. What is a Linear Inequality?

   A linear inequality in one variable is a mathematical statement comparing two linear expressions using an inequality symbol rather than an equality sign. Unlike a linear equation, which typically has a single distinct solution, a linear inequality usually has infinitely many solutions forming a continuous set of numbers.

   Inequality Symbols

   • Less than ($<$): $a < b$ indicates that $a$ lies to the left of $b$ on a standard horizontal number line.
   • Greater than ($>$): $a > b$ indicates that $a$ lies to the right of $b$.
   • Less than or equal to ($\le$): $a \le b$ means that $a$ is either strictly less than $b$ or equal to $b$.
   • Greater than or equal to ($\ge$): $a \ge b$ means that $a$ is either strictly greater than $b$ or equal to $b$.

   The symbols $<$ and $>$ represent strict inequalities, whereas $\le$ and $\ge$ represent non-strict (or inclusive) inequalities.

   Core Algebraic Rules

   Solving linear inequalities mirrors solving linear equations, with one crucial distinction regarding negative multipliers and divisors.

   1. Addition and Subtraction Properties

   Adding or subtracting the same real number from both sides preserves the direction of the inequality:

   • If $a < b$, then $a + c < b + c$.
   • If $a < b$, then $a - c < b - c$.

   2. Multiplication and Division Properties

   Applying multiplication or division depends entirely on the sign of the number involved:

   • Positive Multiplier or Divisor ($c > 0$): The direction of the inequality remains unchanged.
     If $a < b$ and $c > 0$, then $ac < bc$ and $\frac{a}{c} < \frac{b}{c}$.
   • Negative Multiplier or Divisor ($c < 0$): The inequality symbol must be reversed.
     If $a < b$ and $c < 0$, then $ac > bc$ and $\frac{a}{c} > \frac{b}{c}$.

   Example: Consider the true statement $-2 < 5$. Multiplying both sides by $-3$ yields $6$ and $-15$. Because $6$ is greater than $-15$, the inequality reverses to $6 > -15$.

   Step-by-Step Solution Procedure

   To solve a multi-step linear inequality, apply systematic algebraic transformations to isolate the variable on one side.

   1. Clear Fractions and Decimals: Multiply every term by the least common denominator if necessary.
   2. Simplify Expressions: Apply the distributive property to expand parentheses, then combine like terms on each side.
   3. Collect Variable Terms: Add or subtract variable terms so that all variables reside on one side.
   4. Isolate the Variable: Add or subtract constant terms, then divide or multiply by the variable's coefficient, reversing the inequality symbol if that coefficient is negative.

   Worked Example

   Solve the inequality: $$3(2x - 4) - 5x \le 4x + 6$$

   Step 1: Expand parentheses and combine like terms on the left side:

   $$6x - 12 - 5x \le 4x + 6$$

   $$x - 12 \le 4x + 6$$

   Step 2: Subtract $4x$ from both sides:

   $$-3x - 12 \le 6$$

   Step 3: Add $12$ to both sides:

   $$-3x \le 18$$

   Step 4: Divide by $-3$ and reverse the inequality sign:

   $$x \ge -6$$

   Notations and Number Line Graphing

   The solution to an inequality can be communicated in three conventional ways: set-builder notation, interval notation, and graphical representation on a number line.

   Interval Notation

   Interval notation uses brackets and parentheses to denote boundary points:

   • Parenthesis $( \text{ or } )$: Indicates that the endpoint is excluded (used with $<$, $>$, and $\pm\infty$).
   • Bracket $[ \text{ or } ]$: Indicates that the endpoint is included in the solution set (used with $\le$ and $\ge$).

   For example, the solution $x \ge -6$ is written in interval notation as $[-6, \infty)$, while $x < 3$ is written as $(-\infty, 3)$.

   Number Line Representation

   • A solid dot (closed circle) at a value denotes an inclusive boundary ($\le$ or $\ge$).
   • An open circle denotes an exclusive boundary ($<$ or $>$).
   • Shading is drawn along the axis in the direction that satisfies the inequality, with an arrow indicating continuation toward $\infty$ or $-\infty$.

   Compound Inequalities

   A compound inequality combines two distinct inequalities using either the logical conjunction AND or the disjunction OR.

   Conjunctions ("AND")

   A double inequality such as $-3 \le 2x + 1 < 7$ represents an intersection of conditions. Operations must be applied to all three parts simultaneously:

   $$-3 - 1 \le 2x < 7 - 1$$

   $$-4 \le 2x < 6$$

   $$-2 \le x < 3$$

   In interval notation, this solution is $[-2, 3)$.

   Disjunctions ("OR")

   A disjunction represents a union of solutions, such as $x + 2 < -1$ or $2x - 3 \ge 5$. Each inequality is solved independently, yielding $x < -3$ or $x \ge 4$, written as $(-\infty, -3) \cup [4, \infty)$.

   Special Solution Sets

   When algebraic simplification eliminates the variable entirely:

   • If the remaining inequality is universally true (e.g., $4 \le 9$), the solution is all real numbers: $(-\infty, \infty)$.
   • If the remaining statement is false (e.g., $-2 > 5$), there is no solution: $\emptyset$.