| Front | Back | | --- | --- | | What is a **linear inequality** in one variable? | A **linear inequality** in one variable is an inequality that can be written in the form $ax + b < 0$, $ax + b \le 0$, $ax + b > 0$, or $ax + b \ge 0$, where $a$ and $b$ are real numbers and $x$ is the variable. Example: $2x - 3 > 5$. | | What is a **solution set** of an inequality? | The **solution set** of an inequality is the set of all values of the variable(s) that make the inequality true. | | How does the solution set of a one-variable inequality usually look on a number line? | For one variable, the solution set is usually an interval or union of intervals on a number line, for example $x < 4$ or $(2, \infty)$. | | How is the solution set of a two-variable inequality represented in the coordinate plane? | For two variables (like $x$ and $y$), the solution set is typically a shaded region in the coordinate plane showing all points that satisfy the inequality. | | What is a **system of inequalities**? | A **system of inequalities** is a set of two or more inequalities that must all be true at the same time. Its solution set is the overlap (intersection) of the individual solution regions. | | What is meant by an **equivalent** inequality? | Two inequalities are **equivalent** if they have exactly the same solution set. An equivalence transformation changes the inequality without changing which values satisfy it. | | According to Theorem 1, what happens if you add or subtract the same real number on both sides of an inequality? | Adding or subtracting the same real number $c$ on both sides of an inequality gives an equivalent inequality. For example, if $a > b$, then $a + c > b + c$ and $a - c > b - c$. | | According to Theorem 2, what happens when you multiply both sides of an inequality by a **positive** number? | If $k > 0$, multiplying both sides of an inequality by $k$ keeps the inequality direction the same. For example, if $a < b$ and $k > 0$, then $ak < bk$. | | According to Theorem 3, what happens when you multiply or divide both sides of an inequality by a **negative** number? | If $k < 0$, multiplying or dividing both sides of an inequality by $k$ reverses the inequality sign. For example, if $a > b$ and $k < 0$, then $ak < bk$. | | What is a **double inequality**? Give an example and its meaning. | A **double inequality** has a variable expression between two inequality signs, such as $-4 < 3x + 2 < 5$. It means both inequalities hold at the same time: $-4 < 3x + 2$ and $3x + 2 < 5$. | | What does the interval notation $(2, \infty)$ represent in terms of an inequality? | The interval $(2, \infty)$ represents all real numbers greater than 2, corresponding to the inequality $x > 2$. | | How do you interpret the inequality $f_1(x) \le f_2(x)$ using graphs of functions? | Graph $y = f_1(x)$ and $y = f_2(x)$. The solution set of $f_1(x) \le f_2(x)$ is all $x$-values where the graph of $f_1$ lies on or below the graph of $f_2$. | | What is a **boundary line** for a linear inequality in two variables? | The **boundary line** is the line obtained by replacing the inequality sign with an equals sign. For example, for $y \le x + 1$, the boundary line is $y = x + 1$, which separates points that satisfy the inequality from those that do not. | | When do you use a **solid** line versus a **dashed** line for a boundary line? | Use a solid line for $\le$ or $\ge$ (points on the line are included). Use a dashed line for $<$ or $>$ (points on the line are not included). | | How do you decide which side of a boundary line to shade for a two-variable inequality? | Choose a test point not on the boundary line (often $(0,0)$ if allowed). Substitute it into the inequality. If it satisfies the inequality, shade the region containing that point; otherwise shade the opposite side. | | Geometrically, what is the solution set of a **system** of linear inequalities in two variables? | It is the overlap (intersection) of all the individual shaded regions. This region contains all points that satisfy every inequality in the system simultaneously. | | What does it mean if a system of inequalities in one variable has **no common solution**? | It means there is no value of the variable that satisfies all inequalities at once. For example, $x < 1$ and $x \ge 2$ have no common solution, since no number is both less than 1 and at least 2. | | What is the **feasible region** in linear programming? | The **feasible region** is the region in the coordinate plane that contains all points satisfying all the constraint inequalities of a linear programming problem. | | In linear programming, what is the **objective function** and where does an optimal solution occur (if it exists)? | The **objective function** is the linear function representing the quantity to be maximized or minimized (such as profit or cost). If an optimal solution exists, it occurs at one or more vertices (corner points) of the feasible region polygon. | | List the main steps for solving a real-life problem using **linear programming**. | 1. Identify all variables and parameters. 2. Write all **constraints** as linear inequalities. 3. Graph the constraints and find the feasible region. 4. Write the **objective function** in terms of the variables. 5. Evaluate the objective function at each vertex of the feasible region. 6. Choose the vertex that gives the required maximum or minimum value. |

Advanced Number and Bounds3 subtopics

Inequalities and Modelling1 subtopic

Function Transformations4 subtopics

Advanced Probability1 subtopic

Geometric Transformations1 subtopic

Variation and Standard Deviation3 subtopics

Trigonometric Functions and Rules4 subtopics

Sequences, Rational Expressions and Equations5 subtopics

Covariance and Correlation3 subtopics

Logarithms and Non-Linear Inequalities3 subtopics

Graph Theory and Vectors4 subtopics

MYP Extended Mathematics Systems of Inequalities Flashcards