| Front | Back | | --- | --- | | What is a **non-linear inequality in two variables**? | A **non-linear inequality in two variables** is an inequality involving $x$ and $y$ whose boundary is a non-linear curve (such as a parabola, exponential, logarithmic, or rational curve). Its solution is a set of points - a shaded region - in the coordinate plane. | | What does a two-variable inequality like $y < x^2 - 2x - 3$ represent geometrically? | It represents all ordered pairs $(x,y)$ that make the inequality true. Geometrically, it is a shaded region in the plane, not just a single curve. | | In the inequality $y < x^2 - 2x - 3$, what is the **boundary curve**? | The boundary curve is the related equation $y = x^2 - 2x - 3$. | | For $y < x^2 - 2x - 3$, which points satisfy the inequality: those above or below the boundary curve? | Points **below** the curve $y = x^2 - 2x - 3$ satisfy $y < x^2 - 2x - 3$, and points above the curve do not. | | What is a **strict inequality** and how is its boundary drawn? | A **strict inequality** uses $<$ or $>$. Its boundary is not included in the solution set and is drawn as a **dashed** curve. | | What is a **non-strict inequality** and how is its boundary drawn? | A **non-strict inequality** uses $\\le$ or $\\ge$. Its boundary is included in the solution set and is drawn as a **solid** curve. | | Why does the style of the boundary (solid or dashed) matter on a graph? | The boundary style shows **inclusion**: a solid boundary means boundary points are included, while a dashed boundary means boundary points are excluded from the solution set. | | In an inequality of the form $y < f(x)$, on which side of the boundary curve is the solution usually found? | For $y < f(x)$, the solution region is typically the region **below** the boundary curve $y = f(x)$. | | In an inequality of the form $y > f(x)$, on which side of the boundary curve is the solution usually found? | For $y > f(x)$, the solution region is typically the region **above** the boundary curve $y = f(x)$. | | What are the three main steps of the **graph-test-shade** method to solve $y < f(x)$ or $y \\ge f(x)$? | 1. Graph the boundary curve given by $y = f(x)$. 2. Choose a test point not on the boundary and substitute it into the inequality. 3. If the inequality is true at that point, shade the region containing it; if false, shade the opposite region. | | Why is it valid to use a **single test point** to decide which side of a boundary to shade? | A boundary curve splits the plane into separate regions. Within any given region, the inequality is either true for all points or false for all points. So testing a single point in that region tells you whether that entire region is part of the solution. | | For the quadratic inequality $y < -x^2 + 7x - 10$, why is the boundary parabola drawn as dashed? | Because the inequality is strict ($<$), the boundary points are not included, so the parabola $y = -x^2 + 7x - 10$ is drawn as a **dashed** curve. | | For the inequality $y > \\ln(x-2) + 3$, what is the domain restriction on $x$ and why? | The boundary $y = \\ln(x-2) + 3$ is defined only when $x - 2 > 0$, so the domain is $x > 2$. You cannot shade any region where $x \\le 2$ because the logarithmic expression is undefined there. | | For the inequality $y \\le e^{x+1} - 3$, how is the boundary drawn and which side is shaded? | The boundary $y = e^{x+1} - 3$ is drawn as a **solid** curve (because of $\\le$), and the solution region is **on or below** the curve. | | For the inequality $y < \\dfrac{x+3}{x-3}$, what special feature does the boundary have at $x = 3$ and how is it drawn? | At $x = 3$ the expression is undefined, giving a **vertical asymptote** at $x = 3$. This vertical line is always drawn as a **dashed** line because the function is not defined there, regardless of the inequality symbol. | | How does a vertical asymptote in a **rational inequality** affect the solution regions? | A vertical asymptote splits the plane into separate regions (for example, one with $x < 3$ and one with $x > 3$). Each region must be tested separately to determine where the inequality is true, since the function is undefined on the asymptote itself. | | What is a **system of inequalities** in two variables? | A **system of inequalities** is a set of two or more inequalities in the same variables that must all be true simultaneously. A point is a solution only if it satisfies every inequality in the system. | | What is the **feasible region** for a system of inequalities? | The **feasible region** is the set of all points that satisfy every inequality in the system. Graphically, it is the intersection (overlap) of the shaded regions for each inequality, often forming a polygonal region. | | When given a shaded region and its boundary curve, how can you decide whether to use $<$ or $\\le$ (or $>$ or $\\ge$) in the inequality? | Look at the boundary: if it is **dashed**, use a strict inequality ($<$ or $>$). If it is **solid**, use a non-strict inequality ($\\le$ or $\\ge$), since the boundary points are included. | | If a dashed parabola has shading below it, what is the general form of the inequality, and how would this change if the parabola were solid with shading above? | - Dashed boundary with shading below: the inequality is of the form $y < f(x)$. - Solid boundary with shading above: the inequality is of the form $y \\ge f(x)$. |

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 Non-linear Inequalities Flashcards