2.6 Linear Inequalities in One Variable
Key Takeaways
- Single-variable linear inequalities are solved using standard inverse operations, with the critical rule that multiplying or dividing both sides by a negative number reverses the direction of the inequality sign.
- Strict inequalities (<, >) are plotted with open circles on number lines and represented with round parentheses (, ) in interval notation; non-strict inequalities (<=, >=) use solid dots and square brackets [, ].
- A compound 'AND' inequality (a < bx + c <= d) defines the intersection of two conditions and is solved by performing inverse operations simultaneously across all three segments.
- A compound 'OR' inequality defines the union of two disjoint solution sets; a number is a solution if it satisfies at least one of the component inequalities.
- Absolute value inequalities follow two core patterns: |u| < c forms a bounded 'sandwich' -c < u < c, whereas |u| > c splits into two unbounded branches u > c or u < -c.
Linear Inequalities in One Variable
A linear inequality in one variable defines a continuous range of numbers that satisfy a given mathematical constraint. Unlike linear equations, which typically produce a single discrete solution, linear inequalities produce solution sets containing an infinite number of real values.
1. Fundamentals of Inequalities & The Negative Reversal Rule
Linear inequalities use four relational comparison symbols:
- $<$ : strictly less than
- $\le$ : less than or equal to (at most, no more than)
- $>$ : strictly greater than
- $\ge$ : greater than or equal to (at least, no less than)
The Negative Reversal Axiom
When adding or subtracting any quantity, the inequality symbol remains unchanged. However, multiplying or dividing both sides by a negative number reverses (flips) the inequality sign.
+-----------------------------------------------------------------------------+
| WHY NEGATIVE MULTIPLICATION REVERSES |
| |
| Consider the true statement on the number line: |
| 2 < 5 |
| |
| Multiply both sides by -1: |
| -2 > -5 |
| |
| Number Line Visual: |
| <-------+-------+-------+-------+-------+-------+-------+-------+-----> |
| -5 -4 -3 -2 -1 0 1 2 3 4 5 |
| ^ ^ ^ ^ |
| (-5 is further left) (-2 is greater) (2 is <) (5 is >)|
+-----------------------------------------------------------------------------+
Worked Example: Multi-Step Inequality with Sign Reversal
Problem: Solve the inequality for $x$:
Step 1: Distribute the negative constant
Step 2: Combine like terms
Step 3: Collect variable terms on the left (subtract $5x$)
Step 4: Collect constants on the right (subtract 19)
Step 5: Divide by $-13$ and REVERSE the inequality sign
2. Notation Conventions: Number Lines & Interval Notation
The Digital SAT expects fluency in converting between inequality statements, graphical number lines, and formal interval notation.
+-----------------------------------------------------------------------------+
| NOTATION CONVENTIONS GUIDE |
| |
| SYMBOL MEANING NUMBER LINE GRAPH INTERVAL NOTATION |
| ------ ------- ----------------- ----------------- |
| < , > Strict Open Circle ( O ) Round Parentheses ( , ) |
| <= , >= Inclusive Solid Dot ( * ) Square Brackets [ , ] |
| -inf, inf Unbounded Arrow Extends Always Parentheses ( , ) |
+-----------------------------------------------------------------------------+
Comprehensive Notation Conversion Table
| Inequality Statement | Verbal Description | Number Line Visualization | Interval Notation |
|---|---|---|---|
| $x > 3$ | All numbers strictly greater than 3 | Open circle at 3, shaded to the right | $(3, \infty)$ |
| $x \le -2$ | All numbers less than or equal to -2 | Solid dot at -2, shaded to the left | $(-\infty, -2]$ |
| $-1 < x \le 5$ | Numbers between -1 and 5 (excluding -1, including 5) | Open circle at -1, solid dot at 5, shaded between | $(-1, 5]$ |
| $x \le 0 \text{ or } x \ge 4$ | Numbers at most 0 or at least 4 | Solid dot at 0 shaded left, solid dot at 4 shaded right | $(-\infty, 0] \cup [4, \infty)$ |
3. Compound Inequalities: "AND" (Intersections) vs. "OR" (Unions)
A compound inequality combines two individual inequalities using either the conjunction "AND" or "OR".
+-----------------------------------------------------------------------------+
| COMPOUND INEQUALITIES: "AND" VS "OR" |
| |
| "AND" (Intersection: Overlap) "OR" (Union: Combination) |
| - Statement: a <= x <= b - Statement: x < a or x >= b|
| - Must satisfy BOTH conditions - Satisfies EITHER condition|
| - Graph: Single bounded line segment - Graph: Two divergent rays |
| |
| *============* <======* *======> |
| a b a b |
+-----------------------------------------------------------------------------+
Solving Three-Part Compound "AND" Inequalities
To solve a three-part inequality, perform identical operations simultaneously across all three segments:
Add $3$ to all three parts:
Divide all three parts by $4$:
In interval notation, the solution set is $[-2, 5)$.
4. Absolute Value Inequalities: The Two Rules
Absolute value inequalities translate into compound inequalities according to their relational operator:
+-----------------------------------------------------------------------------+
| ABSOLUTE VALUE INEQUALITY RULES |
| |
| RULE 1: "LESS THAN IS SANDWICH (AND)" |
| |u| < c <===> -c < u < c (Bounded segment) |
| |u| <= c <===> -c <= u <= c |
| |
| RULE 2: "GREATER THAN IS SPLIT (OR)" |
| |u| > c <===> u > c OR u < -c (Two separate rays) |
| |u| >= c <===> u >= c OR u <= -c |
+-----------------------------------------------------------------------------+
Worked Example: Absolute Value Inequality
Problem: Solve $|3x - 5| \le 16$ for all real values of $x$.
Step 1: Apply the "Less than Sandwich" rule
Step 2: Add 5 to all three parts
Step 3: Divide by 3
In interval notation, the solution set is $\left[-\frac{11}{3}, 7\right]$.
5. Integer Bounds in SAT Inequality Questions
A favorite SAT question structure asks for the greatest integer or least integer that satisfies a given inequality:
- Multiply both sides by 4: $7 - 3x > 20$
- Subtract 7: $-3x > 13$
- Divide by $-3$ and flip: $x < -\frac{13}{3} \approx -4.333$
- The integers strictly less than $-4.333$ are ${-5, -6, -7, \dots}$.
- The greatest integer in this set is $-5$.
6. Digital SAT Desmos Workflows for Single-Variable Inequalities
In Desmos, you can type single-variable inequalities directly:
- Type
(5 - 2x)/3 > 4into line 1. - Desmos will shade the entire half-plane representing the solution interval ($x < -3.5$).
- The boundary line appears dashed at $x = -3.5$. Click on the boundary line to view its exact $x$-intercept coordinate, making finding integer boundaries effortless.
What is the greatest integer value of x that satisfies the inequality (5 - 2x)/3 > 4?
Which of the following intervals represents the complete solution set to the compound inequality -11 <= 4x - 3 < 17?
A student needs an average score of at least 88 across 4 exams to earn an A- in a course. On the first three exams, the student scored 82, 91, and 85. What is the minimum score the student must achieve on the fourth exam to earn an A-?