When is a boundary line dashed?

For a strict inequality, < or : points on the line are not included.

When is a boundary line solid?

For \le or \ge: points on the line are included.

What is a test point and how is it used?

A point not on the line; substitute it, and if the inequality is true that side is the region.

Why can't (0,0) be a test point for y<3x?

It lies on the line y=3x; use a point such as (1,0) instead.

How do you find a vertex of a region?

Solve the equations of the two boundary lines simultaneously.

Is a vertex on a dashed line part of the region?

No, points on a dashed boundary are not in the region.

Describe the region yx^2-4x+3.

All points above the \cup-shaped curve (inside the bowl), with a dashed boundary.

How do you solve f(x)<0 from the graph of y=f(x)?

Find the x-values where the graph is below the x-axis.

How do you solve f(x)g(x) from two graphs?

Find the x-values where the graph of f is above the graph of g, using the intersection points.

Solve x^2-2x-8<0 from a sketch.

The \cup-curve crosses the axis at -2 and 4 and is below it between them: -2<x<4.

How do you count integer points in a region?

For each integer x, count the integer y values allowed, including boundaries only if solid.

Inequalities represented graphicallyAQA A-Level Maths: Flashcards

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Which side of $y=x+1$ is the region $y>x+1$?

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Which side of $y=x+1$ is the region $y>x+1$ ?: Above the line (strictly, so the boundary is dashed).
When is a boundary line dashed?: For a strict inequality, $<$ or $>$ : points on the line are not included.
When is a boundary line solid?: For $\le$ or $\ge$ : points on the line are included.
How do you describe the region to the right of $x=3$ ?: $x>3$
What is a test point and how is it used?: A point not on the line; substitute it, and if the inequality is true that side is the region.
Why can't $(0,0)$ be a test point for $y<3x$ ?: It lies on the line $y=3x$ ; use a point such as $(1,0)$ instead.
How do you find a vertex of a region?: Solve the equations of the two boundary lines simultaneously.
Is a vertex on a dashed line part of the region?: No, points on a dashed boundary are not in the region.
Describe the region $y>x^2-4x+3$ .: All points above the $\cup$ -shaped curve (inside the bowl), with a dashed boundary.
Region above the curve $C$ and below the line $L$ : write it as inequalities.: $y>$ (curve) and $y<$ (line).
How do you solve $f(x)<0$ from the graph of $y=f(x)$ ?: Find the $x$ -values where the graph is below the $x$ -axis.
How do you solve $f(x)>g(x)$ from two graphs?: Find the $x$ -values where the graph of $f$ is above the graph of $g$ , using the intersection points.
Solve $x^2-2x-8<0$ from a sketch.: The $\cup$ -curve crosses the axis at $-2$ and $4$ and is below it between them: $-2<x<4$ .
How do you count integer points in a region?: For each integer $x$ , count the integer $y$ values allowed, including boundaries only if solid.

Exam questions on Inequalities represented graphically

A region $R$ of the $xy$ -plane is defined by the inequalities $y>x+1$ and $y\le5-x$ .
Find the coordinates of the vertex of $R$ , and state whether this vertex belongs to $R$ .2 marks
The curve $C$ has equation $y=x^2-4x+3$ and the line $L$ has equation $y=x-1$ .
Find the set of values of $x$ for which $C$ lies on or above $L$ .2 marks
A baker plans a day's work using $x$ trays of rolls and $y$ trays of cakes, where $x$ and $y$ are whole numbers. The constraints are $x\ge1$ , $y\ge2$ and $2x+y\le10$ .
The inequalities define a triangular region when drawn. Find the coordinates of its three vertices.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).