Chapter 6 — Linear Inequalities

What is an inequality? An inequality is a mathematical statement that compares two quantities and shows that they are not necessarily equal. Common inequality symbols are:

• < (less than)
• > (greater than)
• ≤ (less than or equal to)
• ≥ (greater than or equal to)

For example, x < 5 means x is any number strictly smaller than 5; x ≤ 5 means x may be 5 or any number less than 5.

Types: Strict inequalities (<, >) exclude the boundary point; non-strict (≤, ≥) include it.

Solution set: The set of all values that satisfy an inequality is its solution set. We often express it as a set, on a number line, or with interval notation.

Basic idea for linear inequalities (one variable): An inequality of the form ax + b < 0 (or >, ≤, ≥) is solved much like a linear equation: isolate x by adding/subtracting and dividing. Key difference: when you multiply or divide both sides by a negative number, the direction of the inequality reverses.

Key properties and operations:

• Addition/Subtraction: if a < b then a + c < b + c (for any real c).
• Multiplication/Division by positive: if a < b and k > 0 then ka < kb.
• Multiplication/Division by negative: if a < b and k < 0 then ka > kb (inequality reverses).
• Transitivity: if a < b and b < c then a < c.

Representation: On the number line use an open dot for strict inequalities and a closed dot for non-strict, shading to the left for < or ≤ and to the right for > or ≥. Interval notation: (a, b), [a, b], (a, b], [a, b) according to open/closed endpoints.

Extension to two variables: A linear inequality in x and y, e.g. ax + by + c < 0, represents a half-plane. The boundary line ax + by + c = 0 is drawn dashed for strict inequalities and solid for non-strict ones; then shade the side that satisfies the inequality (test one point to decide).

Why inequalities matter: They model limits, constraints and feasible ranges in real life: budgets, speed limits, tolerances, thresholds, optimization constraints, etc.

An inequality compares two expressions using symbols >, <, ≥, ≤. The properties of inequalities are rules that let us transform inequalities while keeping track of whether the relation stays the same or reverses. These rules are essential to solve and manipulate linear inequalities in one or two variables.

• Transitive property: If a > b and b > c, then a > c. (Similarly for ≥.)
• Add/subtract the same number: For any real c, if a > b then a + c > b + c and a - c > b - c. Adding or subtracting preserves the inequality.
• Multiply/divide by a positive number: If c > 0 and a > b then ac > bc and a/c > b/c. Multiplying or dividing by a positive number preserves the direction.
• Multiply/divide by a negative number (sign flip): If c < 0 and a > b then ac < bc and a/c < b/c. Multiplying or dividing by a negative number reverses the inequality sign.
• Addition of inequalities: If a > b and c > d then (a + c) > (b + d).
• Reciprocals (positive numbers): If 0 < a < b then 1/a > 1/b (the inequality reverses because the reciprocal function is decreasing on (0, ∞)). For negative a, b, similar care with sign applies.
• Multiplication of positive intervals: If 0 < a < b and 0 < c < d then ac < bd.
• Monotone functions: If f is increasing, a < b ⇒ f(a) < f(b). If f is decreasing, a < b ⇒ f(a) > f(b). This explains effects like squaring (increasing on [0,∞)) and reciprocals (decreasing on (0,∞)).
• Care with non‑known sign: When multiplying or dividing by an expression whose sign is unknown, you must determine its sign first; otherwise the inequality direction may be incorrectly handled.

These properties apply both to one-variable inequalities and, with geometric interpretation, to linear inequalities in two variables (which represent half-planes). They are the backbone of solving, combining, and graphing inequalities.

Definition: A linear inequality in one variable is an inequality that can be written in the form ax + b < 0, ax + b > 0, ax + b <= 0 or ax + b >= 0, where a and b are real numbers and a ≠ 0. The solution is the set of all real numbers x that satisfy the inequality.

General method to solve:

• Isolate the variable on one side by using addition/subtraction and multiplication/division.
• When you add or subtract the same number from both sides, the inequality sign does not change.
• When you multiply or divide both sides by a positive number, the inequality sign does not change. When you multiply or divide by a negative number, the inequality sign reverses direction.
• Express the solution on a number line and/or in interval notation. For strict inequalities (< or >) use open endpoints; for inclusive (<= or >=) use closed endpoints.

Compound inequalities: You may see expressions like a <= bx + c <= d. Treat them like two inequalities joined by AND (intersection). OR situations lead to unions of solution sets.

Verification: Check at least one test value from the solution set and one from its complement to ensure correctness.

Graphical idea: A linear inequality in one variable is represented on the real number line. Solutions form rays or intervals: shade to the left or right of a boundary point; use an open circle for < or > and a filled circle for <= or >=.

What is a compound inequality? A compound inequality joins two or more simple inequalities connected by the words 'and' or 'or'. It describes a set of values of one variable that satisfy both (AND) or at least one (OR) of the inequalities.

Types

• Conjunction (AND): both conditions must hold. Example: a < x ≤ b means x > a and x ≤ b. The solution is the intersection of the solution sets.
• Disjunction (OR): at least one condition must hold. Example: x < c or x ≥ d. The solution is the union of the solution sets.
• Chained inequalities: a < b < c may be used to write simultaneous relations, e.g. a < x ≤ b is shorthand for (x > a) and (x ≤ b).

How to solve

1. For a chained inequality like A < expression ≤ B, treat it as two inequalities linked by AND and perform the same algebraic operations on all three parts simultaneously (add/subtract, multiply/divide). Remember to flip the inequality sign if you multiply or divide by a negative number.
2. For an OR compound inequality, solve each inequality separately, then take the union of the solution intervals.
3. Always express final answers in interval notation and/or with a number-line graph, using open circles for strict inequalities (< or >) and closed dots for inclusive (≤ or ≥).

Important rules

• If you add or subtract the same number from all parts of an inequality, the inequality relations do not change.
• If you multiply or divide all parts by a positive number, the inequality relations do not change. If you multiply or divide by a negative number, you must reverse all inequality signs.
• Chained inequality a < x ≤ b is equivalent to (x > a) ∩ (x ≤ b).

Representation: Use interval notation (a, b), [a, b], (a, b], [a, b) and number-line graphs. For OR, write unions: (-∞, p] ∪ (q, ∞), etc.

What is being described: Interval and set‑builder notations are compact ways to describe sets of real numbers, especially solutions of linear inequalities.

Set‑builder notation: Write a set by a rule. General form: {x ∈ R : condition on x}. Example: {x ∈ R : x > 2} means all real numbers x such that x > 2.

Interval notation: Describes all numbers between endpoints. Brackets [ ] mean endpoint included (≤ or ≥). Parentheses ( ) mean endpoint excluded (< or >). Examples:

• (a, b) = {x ∈ R : a < x < b}
• [a, b] = {x ∈ R : a ≤ x ≤ b}
• [a, b) = {x ∈ R : a ≤ x < b}
• (-∞, b), (a, ∞), (-∞, ∞) for unbounded intervals. Note: ∞ and -∞ are never included, so always use parentheses with them.

How inequalities correspond: Use interval or set‑builder form to represent solution sets of inequalities:

• x > a ↔ (a, ∞) ↔ {x ∈ R : x > a}
• x ≥ a ↔ [a, ∞) ↔ {x ∈ R : x ≥ a}
• a < x ≤ b ↔ (a, b] ↔ {x ∈ R : a < x ≤ b}

Operations on intervals: Intersection and union are used to combine conditions:

• Intersection: numbers satisfying both conditions. Example: [a,b] ∩ [c,d] = [max(a,c), min(b,d)] if the result is nonempty.
• Union: numbers satisfying at least one condition. Example: disjoint intervals (-∞, -1) ∪ (1, ∞).
• Complement: R \ (a,b) = (-∞, a] ∪ [b, ∞).

Endpoints and open/closed circles on number line: On a number line draw a filled (solid) dot for included endpoints (bracket) and an open circle for excluded endpoints (parenthesis). Shade between endpoints or to infinity as required.

Why it matters: Interval and set‑builder notations make writing and combining solution sets of linear inequalities precise and compact. They are used in real problems such as tolerance ranges, eligibility ranges, and constraints.

What it means: Graphical representation on a number line shows the set of real numbers that satisfy a given linear inequality. Each solution set is drawn as a region on a horizontal line with marked boundary points.

Basic rules

• Represent each boundary value (solution of equality) as a point on the line.
• Use a closed dot (●) at a boundary if the inequality includes equality (≤ or ≥).
• Use an open dot (○) at a boundary if the inequality is strict (< or >).
• Shade the line to the right of a boundary for > or ≥ (larger numbers); shade to the left for < or ≤ (smaller numbers).
• If the solution extends indefinitely, draw an arrow (→ or ←) to indicate extension toward +∞ or −∞.

Steps to graph a linear inequality

1. Algebraically solve the inequality for x (isolate x). If you multiply or divide by a negative number, reverse the inequality sign.
2. Mark the boundary point(s) on the number line.
3. Decide open/closed dot depending on strictness.
4. Test one sample point (optional) to confirm which side to shade, then shade the correct region; add arrows for infinite directions.

Compound inequalities and set operations: Use intersection (AND) when both conditions must hold (e.g., 2 ≤ x < 5 is the overlap: closed dot at 2, open at 5, shade between). Use union (OR) when either condition suffices (e.g., x < −1 or x ≥ 4 gives two disjoint shaded regions).

Interval notation corresponds to graphing: (a,b) for a < x < b (open ends), [a,b] for a ≤ x ≤ b (closed ends), (−∞, b], [a, ∞), etc.

Definition: A linear inequality in two variables x and y is an inequality of the form ax + by + c < 0, ax + by + c > 0, ax + by + c ≤ 0 or ax + by + c ≥ 0 where a and b are not both zero. Its solution set is a region (half-plane) in the xy‑plane, not just points.

Boundary and solution region: Replace the inequality by the corresponding equality ax + by + c = 0. This straight line is the boundary of the solution region. If the inequality is strict (< or >) the boundary is not included (draw a dashed line). If it is non‑strict (≤ or ≥) the boundary is included (draw a solid line). The solution set is the half‑plane on one side of the boundary line.

Slope‑intercept form and intercepts: The boundary line can be written as y = (-a/b)x + (-c/b) (when b ≠ 0). The slope is m = -a/b and the y‑intercept is -c/b. x‑intercept is -c/a when a ≠ 0.

Graphing method (step by step):

• 1. Write the boundary: ax + by + c = 0.
• 2. Draw the boundary line (solid for ≤, ≥; dashed for <, >).
• 3. Choose a test point not on the line (commonly (0,0) if it is not on the boundary). Substitute into the inequality.
• 4. If the test point satisfies the inequality, shade the side of the boundary containing that point; otherwise shade the opposite side.

Systems of linear inequalities: The solution to a system of linear inequalities is the intersection of the individual solution regions (often a polygonal region). This is the basic feasible region used in linear programming.

Notes: Always label axes, show intercepts or two points used to draw the boundary, and indicate whether the boundary is included. For strict inequalities use dashed lines; for nonstrict use solid lines.

What it is: A graphical solution of a linear inequality (or a system of linear inequalities) in two variables finds the set of all points (x, y) in the plane that satisfy the inequality. Each linear inequality divides the plane into two half-planes; the solution is the half-plane (or intersection of half-planes) that satisfies the inequality.

General form: ax + by + c < 0, ax + by + c <= 0, ax + by + c > 0 or ax + by + c >= 0 (where a and b are not both zero).

Steps to graph a single linear inequality:

• 1. Replace the inequality by the equality ax + by + c = 0 and draw this straight line (the boundary).
• 2. If the inequality is < or > (strict), draw the boundary as a dashed line. If it is ≤ or ≥ (non-strict), draw it as a solid line.
• 3. Pick a test point that is not on the boundary (commonly (0,0) if it is not on the line). Substitute into the inequality. If the inequality is true at the test point, shade the side of the line containing the test point; otherwise shade the opposite side.

System of linear inequalities: Graph each inequality as above. The solution of the system is the intersection (common shaded region) of all individual solution regions. This common region can be:

• empty (no solution),
• a bounded polygon (when inequalities form a closed region), or
• an unbounded region extending to infinity.

Useful remarks:

• To convert to slope-intercept form: if b ≠ 0, ax + by + c ≤ 0 ⟹ y ≤ (-a/b)x - c/b. This makes plotting easier (y-intercept and slope).
• Intersection points (vertices) of boundary lines are found by solving the corresponding equalities (two linear equations). Solve by substitution, elimination, or determinants (Cramer's rule).
• For optimization problems (linear programming), the optimal value occurs at a vertex (corner point) of the feasible polygonal region.
• Special cases: 0·x + 0·y ≤ k: if k ≥ 0 the solution is the whole plane; if k < 0 the solution is empty.

How to check intersection algebraically: For two boundary lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, their intersection exists and is unique if the determinant D = a1b2 - a2b1 ≠ 0. If D = 0, lines are parallel (either coincident or distinct).

What it is: A system of linear inequalities in two variables is a set of two or more inequalities each of the form ax + by < c, ax + by > c, ax + by ≤ c or ax + by ≥ c (a and b not both zero). The solution of the system is the set of all ordered pairs (x, y) that satisfy every inequality in the system simultaneously.

How to understand it:

• Each inequality divides the plane into two half-planes: the points that satisfy the inequality and the points that do not.
• The boundary of each half-plane is the straight line given by replacing the inequality with equality (ax + by = c). If the inequality is ≤ or ≥, the boundary line is included (drawn solid); if it is < or >, the boundary line is not included (drawn dashed).
• The solution of the system is the intersection of the individual half-planes — i.e., the region (possibly empty) common to all shaded regions.

Graphical method (summary): 1) Rewrite the boundary line in slope-intercept form if convenient: y = (-a/b)x + c/b (when b ≠ 0). 2) Draw the boundary line (solid for ≤ or ≥, dashed for < or >). 3) Pick a test point (often (0,0) if not on the boundary). If the test point satisfies the inequality, shade the side containing it; otherwise shade the opposite side. 4) Repeat for each inequality and identify the overlapping shaded region as the solution set.

Key properties:

• The solution set of a system of linear inequalities in two variables is a convex region — an intersection of half-planes (could be empty, a polygonal region, a ray, a half-plane, or an unbounded region).
• If you need extreme points (vertices) of a bounded feasible region, they occur at intersections of the boundary lines. Solve the corresponding equalities simultaneously to find vertex coordinates.
• This topic is the basis for linear programming in two variables, where a linear objective function is optimized over the feasible (common) region; optimal values occur at vertex points when the feasible region is bounded.

In inequalities we compare expressions using one of four relation signs: the strict inequalities < and >, and the non-strict (or inclusive) inequalities ≤ and ≥. The essential difference is whether the boundary value where the two sides are equal is allowed as part of the solution set.

One-variable view (number line):

• x < a or x > a are strict: the value x = a is not included in the solution. On a number line show an open circle at a and shade to the left for x < a or to the right for x > a.
• x ≤ a or x ≥ a are non-strict (inclusive): the value x = a is included. On a number line show a closed (filled) circle at a and shade accordingly.

Two-variable (linear inequality) / geometric view:

• A linear inequality ax + by + c < 0 or ax + by + c > 0 defines an open half-plane. Its boundary is the straight line ax + by + c = 0. For the strict case (< or >) the boundary line is not included (draw as dashed).
• For ax + by + c ≤ 0 or ax + by + c ≥ 0 the boundary line is included (draw as solid). The solution is the half-plane on the side where the inequality holds.
• To decide which side to shade, pick any convenient test point (commonly (0,0) if it is not on the boundary). Substitute into ax + by + c; if the inequality is true use the side containing the test point, otherwise shade the opposite side.

Key operational rules that affect boundary inclusion:

• Add/subtract the same number from both sides: inequality direction unchanged.
• Multiply/divide both sides by a positive number: direction unchanged.
• Multiply/divide both sides by a negative number: inequality direction reverses (this is important when solving; boundary inclusion depends only on <, >, ≤, ≥, not on sign changes).

Why boundary matters: In many problems (real constraints, safety limits, legal thresholds) allowing equality changes whether a candidate value is acceptable. Graphically, this is the difference between taking the boundary line/point as part of the shaded solution set (inclusive) or leaving it out (exclusive).

Overview: Techniques and Problem Solving for linear inequalities covers methods to solve inequalities in one variable, compound inequalities, rational inequalities, inequalities with absolute values, and systems of linear inequalities in two variables. The main goal is to transform the inequality into a form where the solution set can be read easily (interval form or graph on a number line/plane).

Basic techniques (one variable)

1. Isolate the variable: use addition/subtraction to bring variable terms to one side.
2. Clear coefficients: multiply or divide by a nonzero number. Remember: if you multiply or divide by a negative number, reverse the inequality sign.
3. Combine like terms and simplify to get x < a, x > b, x ≤ c or x ≥ d.
4. Represent solutions on a number line and in interval notation.

Handling compound inequalities: For chained inequalities like a ≤ bx + c < d, perform the same operation on all three segments simultaneously (e.g., subtract c from all parts), keeping track of sign reversals if you multiply/divide by a negative.

Rational inequalities and sign charts: For expressions of the form P(x)/Q(x) > 0 or < 0,

1. bring all terms to one side and write as single rational expression;
2. find critical points: zeros of numerator and denominator (they partition the number line); denominator zeros are excluded from solution;
3. use a sign chart or test points on each interval to determine where the expression is positive or negative;
4. include zeros of numerator if inequality is ≤ or ≥, exclude denominator zeros.

Absolute-value inequalities: Solve |A(x)| < k as -k < A(x) < k. Solve |A(x)| > k as A(x) < -k or A(x) > k. Then solve resulting linear inequalities.

Systems of linear inequalities (two variables): Convert each inequality to standard form (e.g., y < mx + c or ax + by ≤ c), draw the boundary line (dashed for > or <, solid for ≥ or ≤), pick a test point (commonly (0,0) unless on boundary) to decide which side to shade, and the solution is the intersection (common shaded region) for a system.

Problem-solving tips:

• Always perform the same operation on every part of a chained inequality.
• Keep track of strict (<, >) vs non-strict (≤, ≥) to decide open vs closed endpoints on graphs.
• When clearing denominators, multiply both sides by the square of denominators is sometimes used to avoid sign issues, but easier is to bring to one side and use sign charts.
• Check boundary points and special values (where expressions are undefined).
• For word problems, translate constraints into inequalities, solve, and interpret in context (rounding or integer constraints may apply).

Overview
A linear inequality is like a linear equation but describes a range of values (a region) rather than a single line. The topic "Connections and Applications" emphasizes how linear inequalities link to other parts of mathematics (intervals, absolute values, systems of linear equations, linear programming) and how they model real-life constraints.

Key ideas

• One-variable inequalities (e.g. ax + b < 0) describe intervals on the number line. Use algebraic operations to isolate x, remembering to reverse the inequality when multiplying/dividing by a negative number.
• Compound inequalities use AND (intersection) or OR (union). For example, 2 < x + 1 < 5 is an AND that yields an interval.
• Absolute-value inequalities like |x − a| < r become double inequalities a − r < x < a + r; they describe distances from a point.
• Two-variable linear inequalities (ax + by ≤ c) represent half-planes in the plane. The boundary ax + by = c is a straight line; the inequality picks one side.
• Systems of linear inequalities produce feasible regions given by the intersection of half-planes — often a polygon. This is the geometric base for linear programming (optimize a linear objective subject to linear constraints).

How to graph and interpret a two-variable inequality

1. Convert to the boundary line ax + by = c. Draw it (solid if ≤ or ≥, dashed if < or >).
2. Choose a test point (often (0,0) if not on the line). Substitute into the inequality.
3. If the test point satisfies the inequality, shade the side containing that point; otherwise shade the opposite side.

Connections

• With intervals: solutions of one-variable inequalities are intervals; endpoints depend on strictness (open/closed).
• With absolute value: rewrite absolute inequalities as double inequalities (distance interpretation).
• With systems of linear equations: equalities form the boundary lines for inequalities; intersection points (vertices) of these boundaries are critical in optimization problems.
• With linear programming: constraints are linear inequalities; objective functions attain extremes at vertices of the feasible polygon.

Practical interpretation
Linear inequalities model constraints: limits on resources, thresholds, safety bounds, tolerances. The feasible region shows all possible choices meeting all constraints; checking vertices can give best/worst values for linear objectives.