Linear Inequalities and Graphical Solutions

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Linear Inequalities and Graphical Solutions — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionGraphical Representation of InequalitiesConcept Practice

2 marks~3 minCriterion A

A delivery company charges a base fee of 1 dollar plus 2 dollars per kilogram. A customer's budget satisfies

y \leq 2x + 1

, where

x

is the mass (kg) of the parcel and

y

is the maximum total cost (dollars) the customer will pay.

On the coordinate grid provided, construct the graph of

y \leq 2x + 1

for

0 \leq x \leq 4

, showing clearly the boundary line and the solution region. [1]

State the coordinates of one point in the solution region. Justify whether a 2 kg parcel costing 6 dollars satisfies the customer's budget constraint. [1]

Solutions

2QuestionGraphical Representation of InequalitiesAssessment Practice

2 marks~3 minCriterion D

A student plans a party with a budget of 30 dollars and can carry at most 10 items. Let

x

represent the number of snacks and

y

the number of drinks, where each snack costs 2 dollars and each drink costs 3 dollars. This situation is modelled by the system:

2x + 3y \leq 30, \quad x + y \leq 10, \quad x \geq 0, \quad y \geq 0

Justify whether this system of linear inequalities is a reliable model for the student's party-planning decisions. [2]

Solutions

3QuestionRepresenting Solutions on a Number LineAssessment Practice

4 marks~6 minCriterion C

A building's emergency generator activates when the outdoor temperature

T

(in °C) satisfies the inequality shown on the number line below.

Write the inequality represented by the number line in the form

a \leq T < b

. [1]

Justify whether

T = -2

and

T = 4

each belong to the solution set, referring to the number line notation. [2]

The generator's manual states it must activate "at any temperature strictly below 4°C, including −2°C." Assess whether the inequality you wrote in part (a) fully satisfies this requirement. [1]

Solutions

4QuestionGraphical Representation of InequalitiesAssessment Practice

6 marks~9 minCriterion B

A city planner is designing a pedestrian zone. The safe walking corridor must satisfy two constraints modelled by linear inequalities. Three trial corridor systems are tested:

System 1:

y \geq x + 1

and

y \leq -x + 5

System 2:

y \geq x + 3

and

y \leq -x + 3

System 3:

y \geq x + 5

and

y \leq -x + 1

A proposed corridor uses the system

y \geq x + 2

and

y \leq -x + 4

Analyse the three trial systems. For each, identify whether the feasible region is bounded, a single point, or empty, and explain how the relationship between the constants

c_1

and

c_2

determines this outcome. [2]

Deduce the nature of the feasible region for the proposed corridor system, justifying your answer using the pattern identified in part (a). [1]

Construct a graph of the proposed corridor system, clearly showing the boundary lines, the feasible region, and the coordinates of any intersection points. Hence advise the city planner whether the proposed corridor should be approved, justifying your recommendation with reference to the feasible region. [3]

Solutions

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