Linear Inequalities and Graphical Solutions

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

1QuestionRepresenting Solutions on a Number LineConcept Practice

2 marks~3 minCriterion A

A city regulation states that the noise level in a residential area must be at most 3 units above the baseline.

The number line below represents the permitted noise levels,

x

.

[See diagram]

Write the inequality represented by the number line. [1]

A monitor records a noise level of

x = 3

. Justify whether this reading complies with the regulation, using the inequality. [1]

Solutions

2QuestionInequality Word ProblemsAssessment Practice

2 marks~3 minCriterion C

A structural engineer is designing a triangular brace for a support frame. Two sides of the triangular brace measure 5 cm and 7 cm. The third side has length

x

cm.

Deduce the range of possible values of

x

. [1]

The engineer states: "A third side of 11.5 cm is acceptable, but a third side of 2 cm is not." Justify whether this statement is correct. [1]

Solutions

3QuestionGraphical Representation of InequalitiesAssessment Practice

5 marks~8 minCriterion D

A logistics company models the number of small parcels (

x

) and large parcels (

y

) loaded into a delivery van using:

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

Construct the feasible region for this system of inequalities. Label both axes, all boundary lines, and the feasible region clearly. State the coordinates of all vertices. [2]

The company considers adding the constraint

y \geq 1

. Identify how the feasible region changes and state the new set of vertices. [1]

A dispatcher claims that any combination within the feasible region from part (b) is operationally valid. Advise the dispatcher whether the feasible region alone is sufficient to guarantee valid loading decisions, identifying at least one mathematical limitation and one real-world limitation of the model. [2]

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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