MYP Extended Mathematics Topic Non-linear Inequalities Revision Notes

MYP Extended Mathematics Topic Non-linear Inequalities Revision Notes | RevisionDojo

A Non-Linear Inequality in Two Variables

A non-linear inequality in two variables (usually $x$ and $y$) describes a region of the coordinate plane, not just a single curve.
The "boundary" of that region is the related equation, and whether the boundary is included depends on the inequality symbol.

Definition

Non-linear inequality (in two variables)

An inequality involving $x$ and $y$ whose boundary is a non-linear curve (for example, a parabola, exponential curve, logarithmic curve, or rational curve). Its solution is a set of points (a shaded region) in the plane.

The Solution to a Two-Variable Inequality Is a Region of Points

When you solve an inequality such as $$y < x^2 - 2x - 3$$ you are looking for all ordered pairs $(x,y)$ that make the inequality true.
Graphically:
1. The curve $y = x^2 - 2x - 3$ is the boundary.
2. Points below the curve satisfy $y < x^2 - 2x - 3$.
3. Points above the curve satisfy $y > x^2 - 2x - 3$.

Note

It helps to read inequalities like "$y$ is less than (the function of $x$)."
If $y$ must be less, the solution is typically the region below the boundary curve.

Boundary Curves: Solid vs Dashed

A key graphical idea is whether points on the boundary are included.

Definition

Strict inequality

An inequality using $<$ or $>$, meaning the boundary is not included in the solution set.

Definition

Non-strict inequality

An inequality using $\le$ or $\ge$, meaning the boundary is included in the solution set.

If the inequality is strict (for example, $y < f(x)$ or $y > f(x)$), draw the boundary as a dashed curve.
If the inequality is non-strict (for example, $y \le f(x)$ or $y \ge f(x)$), draw the boundary as a solid curve.

Common Mistake

A very common mistake is drawing a solid boundary for $<$ or $>$, or a dashed boundary for $\le$ or $\ge$.
The boundary style communicates inclusion.

A Reliable Method for Graphing Non-Linear Inequalities

For many non-linear inequalities, you cannot "solve" them into a single interval easily. The standard approach is graphical.

Method: Graph, Test, Shade

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A Non-Linear Inequality in Two Variables

A non-linear inequality in two variables (usually $x$ and $y$) describes a region of the coordinate plane, not just a single curve.
The "boundary" of that region is the related equation, and whether the boundary is included depends on the inequality symbol.

Definition

Non-linear inequality (in two variables)

The Solution to a Two-Variable Inequality Is a Region of Points

When you solve an inequality such as $$y < x^2 - 2x - 3$$ you are looking for all ordered pairs $(x,y)$ that make the inequality true.
Graphically:
1. The curve $y = x^2 - 2x - 3$ is the boundary.
2. Points below the curve satisfy $y < x^2 - 2x - 3$.
3. Points above the curve satisfy $y > x^2 - 2x - 3$.

Note

It helps to read inequalities like "$y$ is less than (the function of $x$)."
If $y$ must be less, the solution is typically the region below the boundary curve.

Boundary Curves: Solid vs Dashed

A key graphical idea is whether points on the boundary are included.

Definition

Strict inequality

An inequality using $<$ or $>$, meaning the boundary is not included in the solution set.

Definition

Non-strict inequality

An inequality using $\le$ or $\ge$, meaning the boundary is included in the solution set.

If the inequality is strict (for example, $y < f(x)$ or $y > f(x)$), draw the boundary as a dashed curve.
If the inequality is non-strict (for example, $y \le f(x)$ or $y \ge f(x)$), draw the boundary as a solid curve.

Common Mistake

A very common mistake is drawing a solid boundary for $<$ or $>$, or a dashed boundary for $\le$ or $\ge$.
The boundary style communicates inclusion.

A Reliable Method for Graphing Non-Linear Inequalities

For many non-linear inequalities, you cannot "solve" them into a single interval easily. The standard approach is graphical.

Method: Graph, Test, Shade

Definition

Paywall

(on a website) an arrangement whereby access is restricted to users who have paid to subscribe to the site.

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Duis aute irure dolor in reprehenderit

Note

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Excepteur sint occaecat cupidatat non proident

Tip

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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 Notes

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

A Non-Linear Inequality in Two Variables

The Solution to a Two-Variable Inequality Is a Region of Points

Boundary Curves: Solid vs Dashed

A Reliable Method for Graphing Non-Linear Inequalities

Method: Graph, Test, Shade

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

A Non-Linear Inequality in Two Variables

The Solution to a Two-Variable Inequality Is a Region of Points

Boundary Curves: Solid vs Dashed

A Reliable Method for Graphing Non-Linear Inequalities

Method: Graph, Test, Shade

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account