First-order inequalities are an important branch of algebra that requires an understanding of the properties of inequalities and the ability to work with intervals. In the YÖS exam, inequality problems appear not only on their own, but also as part of more complex problems. A deep understanding of this topic is critically important for success.
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Definition: An inequality is a relation between two expressions denoted by one of the signs:- $>$ (greater than)- $<$ (less than)- $\geq$ (greater than or equal to)- $\leq$ (less than or equal to)
Examples:- $3x + 2 > 5$ — strict inequality- $2x - 1 \leq 7$ — non-strict inequality
Strict inequalities: Use the signs $>$ or $<$. The boundary is not included in the solution.
Non-strict inequalities: Use the signs $\geq$ or $\leq$. The boundary is included in the solution.
Definition: The solution of an inequality is the set of all values of the variable for which the inequality is true.
Note: The solution of an inequality is usually represented as an interval or a union of intervals.
Property 1 (Reflexivity): $a \geq a$ for all $a$.
Property 2 (Transitivity): If $a > b$ and $b > c$, then $a > c$.
Property 3 (Antisymmetry): If $a \geq b$ and $b \geq a$, then $a = b$.
Addition and subtraction: If $a > b$, then $a + c > b + c$ and $a - c > b - c$ for any $c$.
$$a > b \Rightarrow a + c > b + c$$
Multiplication and division by a positive number: If $a > b$ and $c > 0$, then $ac > bc$ and $\frac{a}{c} > \frac{b}{c}$.
$$a > b, \quad c > 0 \Rightarrow ac > bc$$
Multiplication and division by a negative number: If $a > b$ and $c < 0$, then $ac < bc$ and $\frac{a}{c} < \frac{b}{c}$.
IMPORTANT: When multiplying or dividing by a negative number, the inequality sign is reversed!
$$a > b, \quad c < 0 \Rightarrow ac < bc$$
Example 1: If $x > 3$, then $x + 5 > 8$
Solution:$$x > 3$$$$x + 5 > 3 + 5$$$$x + 5 > 8$$
Example 2: If $2x < 10$, then $x < 5$
Solution:$$2x < 10 \quad | \text{divide by 2 (a positive number)}$$$$x < 5$$
Example 3: If $-x > 4$, then $x < -4$
Solution:$$-x > 4 \quad | \text{multiply by } -1 \text{ (a negative number)}$$$$x < -4 \quad \text{(the sign is reversed!)}$$
Definition: A linear inequality in one variable has the form:$$ax + b > c \quad \text{or} \quad ax + b < c$$$$ax + b \geq c \quad \text{or} \quad ax + b \leq c$$
where $a \neq 0$.
Example 1: Solve $3x - 5 > 7$
Solution:$$3x > 7 + 5$$$$3x > 12$$$$x > 4$$
Answer: $x \in (4, +\infty)$
Example 2: Solve $-2x + 6 \leq 10$
Solution:$$-2x \leq 10 - 6$$$$-2x \leq 4 \quad | \text{divide by } -2$$$$x \geq -2 \quad \text{(the sign is reversed!)}$$
Answer: $x \in [-2, +\infty)$
Example 3: Solve $\frac{x}{2} + 3 > 5$
Solution:$$\frac{x}{2} > 5 - 3$$$$\frac{x}{2} > 2$$$$x > 4$$
Open interval: $(a, b)$ — includes the numbers between $a$ and $b$, but not the endpoints themselves.$$x \in (a, b) \Leftrightarrow a < x < b$$
Closed interval: $[a, b]$ — includes the numbers between $a$ and $b$, including the endpoints.$$x \in [a, b] \Leftrightarrow a \leq x \leq b$$
Half-open intervals:- $[a, b)$: $a \leq x < b$- $(a, b]$: $a < x \leq b$
Infinite intervals:- $(a, +\infty)$: $x > a$- $[a, +\infty)$: $x \geq a$- $(-\infty, b)$: $x < b$- $(-\infty, b]$: $x \leq b$- $(-\infty, +\infty)$: all real numbers
On the number line:- An open circle $($ or $)$ means that the point is not included- A closed circle $[$ or $]$ means that the point is included
Example:```For x > 3:─────●━━━━━━→ 3(point 3 is not included)
For x ≤ 5:←━━━━━━●───── 5(point 5 is included)```
Definition: A double inequality has the form:$$a < x < b \quad \text{or} \quad a \leq x \leq b$$etc. (combinations of strict and non-strict)
Solution: A double inequality can be solved as a system of two inequalities, or by applying operations simultaneously to all parts.
Example 1: Solve $1 < x + 2 < 5$
Solution (simultaneous operations):$$1 - 2 < x < 5 - 2$$$$-1 < x < 3$$
Answer: $x \in (-1, 3)$
Example 2: Solve $-3 \leq 2x - 1 \leq 7$
Solution:$$-3 + 1 \leq 2x \leq 7 + 1$$$$-2 \leq 2x \leq 8$$$$-1 \leq x \leq 4$$
Answer: $x \in [-1, 4]$
Example 3: Solve $0 < \frac{x}{2} + 1 < 3$
Solution:$$0 - 1 < \frac{x}{2} < 3 - 1$$$$-1 < \frac{x}{2} < 2$$$$-2 < x < 4$$
Answer: $x \in (-2, 4)$
Definition: A system of inequalities is a set of inequalities that must all be satisfied simultaneously.
Solution: The intersection of the solution sets of each inequality.
$$\begin{cases}f_1(x) > a_1 \\f_2(x) < a_2\end{cases}$$
Example 1:$$\begin{cases}x > 2 \\x < 5\end{cases}$$
Solution: The intersection of $(2, +\infty)$ and $(-\infty, 5)$ is $(2, 5)$.
Answer: $x \in (2, 5)$
Example 2:$$\begin{cases}2x + 1 > 5 \\3x - 2 \leq 10\end{cases}$$
Solution of the first:$$2x > 4 \Rightarrow x > 2$$
Solution of the second:$$3x \leq 12 \Rightarrow x \leq 4$$
Intersection: $x \in (2, 4]$
Example 3:$$\begin{cases}x + 1 > 3 \\x - 2 < 0\end{cases}$$
Solution of the first: $x > 2$Solution of the second: $x < 2$
Intersection: $\emptyset$ (empty, no solutions)
Definition: A collection of inequalities is a set of inequalities of which at least one must be satisfied.
Solution: The union of the solution sets of each inequality.
$$\left[\begin{array}{l}f_1(x) > a_1 \\f_2(x) < a_2\end{array}\right.$$
(The square bracket on the left indicates a collection, not a system)
Example 1:$$\left[\begin{array}{l}x < 1 \\x > 3\end{array}\right.$$
Solution: The union of $(-\infty, 1)$ and $(3, +\infty)$.
Answer: $x \in (-\infty, 1) \cup (3, +\infty)$
Example 2:$$\left[\begin{array}{l}2x - 1 < 3 \\3x + 2 > 8\end{array}\right.$$
Solution of the first:$$2x < 4 \Rightarrow x < 2$$
Solution of the second:$$3x > 6 \Rightarrow x > 2$$
Union: $(-\infty, 2) \cup (2, +\infty) = (-\infty, +\infty) \setminus \{2\}$, or simply all numbers except 2.
Problem: Find all integers that satisfy the given inequality.
Example 1: Find the integer solutions of $2x - 3 < 5$
Solution:$$2x < 8$$$$x < 4$$
Integers less than 4: $\ldots, 1, 2, 3$
If positive integers are required: $1, 2, 3$
Example 2: Find the integer solutions of $-1 < x \leq 3$
Solution: The inequality is already solved.
Integers in the interval $(-1, 3]$: $0, 1, 2, 3$
Example 3: Find the integer solutions of $\frac{x}{2} > -2$ and $x < 1$
Solution of the first: $x > -4$Solution of the second: $x < 1$
Intersection: $-4 < x < 1$
Integers: $-3, -2, -1, 0$
Form 1: $|x| < a$ (where $a > 0$)$$|x| < a \Leftrightarrow -a < x < a$$
Form 2: $|x| > a$ (where $a > 0$)$$|x| > a \Leftrightarrow x < -a \text{ or } x > a$$
Form 3: $|x| \leq a$$$|x| \leq a \Leftrightarrow -a \leq x \leq a$$
Form 4: $|x| \geq a$$$|x| \geq a \Leftrightarrow x \leq -a \text{ or } x \geq a$$
Example 1: Solve $|x| < 3$
Solution:$$-3 < x < 3$$
Answer: $x \in (-3, 3)$
Example 2: Solve $|x - 2| \leq 4$
Solution:$$-4 \leq x - 2 \leq 4$$$$-4 + 2 \leq x \leq 4 + 2$$$$-2 \leq x \leq 6$$
Answer: $x \in [-2, 6]$
Example 3: Solve $|2x + 1| > 5$
Solution:$$2x + 1 < -5 \quad \text{or} \quad 2x + 1 > 5$$$$2x < -6 \quad \text{or} \quad 2x > 4$$$$x < -3 \quad \text{or} \quad x > 2$$
Answer: $x \in (-\infty, -3) \cup (2, +\infty)$
Statement: Solve $5x - 2 > 3x + 6$
Solution:$$5x - 3x > 6 + 2$$$$2x > 8$$$$x > 4$$
Statement:$$\begin{cases}3x + 1 > 10 \\2x - 5 < 7\end{cases}$$
Solution of the first:$$3x > 9 \Rightarrow x > 3$$
Solution of the second:$$2x < 12 \Rightarrow x < 6$$
Intersection: $3 < x < 6$
Answer: $x \in (3, 6)$
Statement: Find all integers satisfying $-2 < x + 1 \leq 4$
Solution:$$-2 - 1 < x \leq 4 - 1$$$$-3 < x \leq 3$$
Integers: $-2, -1, 0, 1, 2, 3$
Answer: 6 integers
Statement: Solve $|x + 2| \leq 5$
Solution:$$-5 \leq x + 2 \leq 5$$$$-7 \leq x \leq 3$$
Answer: $x \in [-7, 3]$
Statement: Solve $-1 < \frac{2x - 3}{2} < 5$
Solution:$$-1 \cdot 2 < 2x - 3 < 5 \cdot 2$$$$-2 < 2x - 3 < 10$$$$-2 + 3 < 2x < 10 + 3$$$$1 < 2x < 13$$$$\frac{1}{2} < x < \frac{13}{2}$$$$0.5 < x < 6.5$$
Answer: $x \in (0.5, 6.5)$
Incorrect: $-x > 5 \Rightarrow x > -5$
Correct: $-x > 5 \Rightarrow x < -5$
Incorrect: $|x + 3| < 2 \Rightarrow x + 3 < 2$ (forgot the second part)
Correct: $|x + 3| < 2 \Leftrightarrow -2 < x + 3 < 2$
Incorrect: Taking the union instead of the intersection for a system.
Correct: A system means intersection; a collection means union.
Incorrect: Shading the region together with the point, even though the inequality was strict.
Correct: Open circle for strict inequalities ($<$ or $>$), closed circle for non-strict ($\leq$ or $\geq$).
Incorrect: Solving $|x| > 3$ only as $x > 3$.
Correct: $|x| > 3 \Leftrightarrow x < -3$ or $x > 3$
Always substitute the boundary value back into the original inequality to check whether it is included in the solution.
A graphical representation helps you visualize the solution and reduces the number of mistakes.
When working with negative numbers, especially when multiplying or dividing by them, it is easy to make a sign error.
Develop your own method for tracking sign changes under various operations.
Finding integer solutions often appears in YÖS, so pay particular attention to it.
If you are unsure, substitute a point from the assumed solution into the original inequality.
First-order inequalities are a basic but important tool in algebra. Understanding the properties of inequalities, being able to work with intervals, solving systems and collections of inequalities, as well as basic skills in working with absolute value — all of these are necessary for successfully solving YÖS problems. The key to success is attentiveness, a systematic approach, and constant practice, especially when working with signs and boundary values.
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