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Equations, Inequalities and Formulae

Equations, Inequalities and Formulae — 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, xx.

[See diagram]
a
Write the inequality represented by the number line. [1]
b
A monitor records a noise level of x=3x = 3. Justify whether this reading complies with the regulation, using the inequality. [1]
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2QuestionChecking Solutions for Both EquationsConcept Practice
2 marks~3 minCriterion A
A city planner models two proposed road alignments on a coordinate grid. Road 1 follows y=2x+1y = 2x + 1 and Road 2 follows y=−x+4y = -x + 4. A junction is planned at point P(1,3)P(1, 3).

Justify that P(1,3)P(1, 3) is the junction point of the two roads by verifying it satisfies both equations. [2]
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3QuestionSolving by FactorizationConcept Practice
4 marks~6 minCriterion A
A landscape architect models the cross-section of a drainage channel using the function y=x2−x−6y = x^2 - x - 6, where xx is the horizontal distance (metres) from a reference point and yy is the height (metres) relative to ground level. The graph is shown, with its xx-intercepts and vertex clearly labelled.
a
Deduce the pair of factors of −6-6 whose product is −6-6 and whose sum is −1-1. [1]
b
Solve x2−x−6=0x^2 - x - 6 = 0 by factorization. [1]
c
The channel walls meet ground level where y=0y = 0. Using your solutions from part (b), calculate the width of the channel at ground level, then advise the architect whether the design meets the requirement that the width must be at least 4 m. [2]
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4QuestionUnderstanding Literal EquationsConcept Practice
2 marks~3 minCriterion A
A community garden is being enclosed with fencing. The garden is rectangular, with length ll and width ww, and its perimeter is given by

P=2l+2w.P = 2l + 2w.

The available fencing is 26 m. The garden measures l=8l = 8 m and w=5w = 5 m.
a
State what PP represents and write down the formula rearranged to make ll the subject. [1]
b
Using the given dimensions, calculate PP and justify whether the available fencing is sufficient to enclose the garden. [1]
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5QuestionMulti-Step Problems with VariablesConcept Practice
2 marks~3 minCriterion B
A community centre arranges daily volunteer shifts. The shift lengths (in hours) follow a pattern of consecutive integers:

Day 1: 1+2=31 + 2 = 3 hours
Day 2: 2+3+4=92 + 3 + 4 = 9 hours
Day 3: 3+4+5+6=183 + 4 + 5 + 6 = 18 hours
Day 4: 4+5+6+7+8=304 + 5 + 6 + 7 + 8 = 30 hours

On Day 5, shifts run for 6 consecutive integers starting at 5.
a
Deduce the pattern connecting the sum of each set to its terms. [1]
b
A volunteer coordinator claims Day 5 totals exactly 45 hours of coverage. Justify whether this claim is correct. [1]

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6QuestionInterpreting and Validating SolutionsConcept Practice
2 marks~3 minCriterion C
A mobile data plan charges a monthly fee. Plan A costs 4x−74x - 7 dollars and Plan B costs 2x+52x + 5 dollars, where xx is the number of gigabytes used.

A customer claims that both plans cost the same when x=6x = 6 gigabytes.

Substitute x=6x = 6 into both expressions, show your working, and justify whether the customer's claim is correct. [2]

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7QuestionChoosing Appropriate MethodsConcept Practice
4 marks~6 minCriterion A
A cyclist's journey is modelled by a straight-line graph of distance (km) against time (hours). The line passes through the points (1,15)(1, 15) and (3,45)(3, 45).
a
Calculate the cyclist's speed. [1]
b
Deduce the equation of the line in the form d=mtd = mt, where dd is distance in km and tt is time in hours. [1]
c
The cyclist's destination is 70 km away. Using your equation, advise whether the cyclist should plan to complete the journey within a 5-hour window. Justify your answer. [2]
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8QuestionSolving One-Step and Two-Step EquationsConcept Practice
4 marks~6 minCriterion A
A mobile phone plan charges a fixed monthly fee plus a cost per gigabyte of data used. Last month, a customer used 7 GB of additional data and paid a total bill of 15 dollars. This situation is modelled by the equation x+7=15x + 7 = 15, where xx is the fixed monthly fee in dollars.
a
Calculate the fixed monthly fee xx. [1]
b
Deduce the operation applied to both sides of the equation to isolate xx, and state the property of equality that justifies this step. [2]
c
Advise the customer, who budgets 10 dollars per month for fixed fees, whether this plan is within their budget. Justify your advice using your answer to part (a). [1]
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9QuestionGraphical Representation of InequalitiesAssessment Practice
5 marks~8 minCriterion D
A logistics company models the number of small parcels (xx) and large parcels (yy) loaded into a delivery van using:

x+y≤10,x≥2,y≥0x + y \leq 10, \quad x \geq 2, \quad y \geq 0
a
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]
b
The company considers adding the constraint y≥1y \geq 1. Identify how the feasible region changes and state the new set of vertices. [1]
c
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]
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10QuestionInequality 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 xx cm.
a
Deduce the range of possible values of xx. [1]
b
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]
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11QuestionGraphical 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≥x+1y \geq x + 1 and y≤−x+5y \leq -x + 5
System 2: y≥x+3y \geq x + 3 and y≤−x+3y \leq -x + 3
System 3: y≥x+5y \geq x + 5 and y≤−x+1y \leq -x + 1

A proposed corridor uses the system y≥x+2y \geq x + 2 and y≤−x+4y \leq -x + 4.
a
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 c1c_1 and c2c_2 determines this outcome. [2]
b
Deduce the nature of the feasible region for the proposed corridor system, justifying your answer using the pattern identified in part (a). [1]
c
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]
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12QuestionChecking Solutions for Both EquationsAssessment Practice
6 marks~9 minCriterion B
A logistics company tracks two delivery variables, xx (vehicles dispatched) and yy (spare vehicles held in reserve), which satisfy:

2x+y=102x + y = 10
x−y=cx - y = c

where cc is a whole-number demand index set each morning.
a
Solve the system for c=2c = 2, c=5c = 5, and c=8c = 8. Present your solutions as:

cc258
(x,y)(x, y)_________ [2]
b
Analyse how xx and yy each change as cc increases by 3. Hence deduce a general rule expressing xx and yy in terms of cc. [2]
c
The company requires at least one spare vehicle (y≥1y \geq 1) to operate safely. Using your general rule, determine the maximum whole-number value of cc for which this condition is met, and advise the operations manager whether the company can operate safely when c=14c = 14. [2]

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13QuestionChecking Solutions for Both EquationsAssessment Practice
4 marks~6 minCriterion D
A city planner models two road-widening proposals. Proposal A follows the cost equation y=x−1y = x - 1 and Proposal B follows y=−x+5y = -x + 5, where xx is the number of months into the project and yy is the projected budget surplus (in millions of dollars). The graph shows both lines intersecting at point PP.
a
State the coordinates of PP from the graph. [1]
b
Show that PP satisfies both equations simultaneously. [2]
c
Interpret what the coordinates of PP mean in this context, and advise the city planner whether PP should be used as the basis for deciding when both proposals are financially equivalent. [1]
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14QuestionSolving GraphicallyAssessment Practice
4 marks~6 minCriterion C
A city planner models two proposed road alignments as straight lines on a coordinate grid. The graph shows both lines.

Line 1 passes through (0,1)(0, 1) and (2,5)(2, 5).
Line 2 passes through (0,−3)(0, -3) and (2,1)(2, 1).
a
Deduce the gradient of each line from the graph. [1]
b
Explain why the two roads will never intersect, using the properties of both lines. [2]
c
Advise the city planner whether a single crossing point between the two roads can ever be achieved by extending them indefinitely, and justify your advice. [1]
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15QuestionPractice Quadratic Equations solvingAssessment Practice
5 marks~8 minCriterion B
A structural engineer models the cross-sectional dimensions of two roof panels using quadratic equations. The solutions of each equation represent the two dimensions (in metres) of a panel.

Equation 1: x2+6x+5=0x^2 + 6x + 5 = 0
Equation 2: x2−8x+12=0x^2 - 8x + 12 = 0
Equation 3: x2+2x−8=0x^2 + 2x - 8 = 0
Equation 4: x2−10x+21=0x^2 - 10x + 21 = 0
a
Solve each equation and calculate the sum of its two solutions. Explain the relationship between each sum and the coefficient of xx. [1]
b
Calculate the product of the two solutions for each equation. Explain the relationship between each product and the constant term. [2]
c
A fifth panel is modelled by x2−7x+10=0x^2 - 7x + 10 = 0. Deduce the sum and product of its solutions without solving the equation, then justify whether this panel could have two positive dimensions. [2]

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16QuestionSolving by FactorizationAssessment Practice
2 marks~3 minCriterion D
A carpenter is designing a rectangular brace for a gate. The brace has two perpendicular sides of length (x+1)(x + 1) cm and (x+2)(x + 2) cm, and a diagonal support of exactly 55 cm.
a
Show that the side lengths satisfy x2+3x−10=0x^2 + 3x - 10 = 0. [1]
b
Solve x2+3x−10=0x^2 + 3x - 10 = 0 by factorisation to find both values of xx. Hence state the two side lengths, and justify which value of xx is valid in this context. [1]
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17QuestionSolving by FactorizationAssessment Practice
4 marks~6 minCriterion C
A landscape architect models the edge of a curved garden bed using the equation y=3x2−12xy = 3x^2 - 12x, where xx is the horizontal distance in metres from a reference post. The graph of this function is shown below.
a
Determine the highest common factor (HCF) of 3x23x^2 and 12x12x. [1]
b
Deduce the roots of 3x2−12x=03x^2 - 12x = 0 by factorization. [1]
c
The architect's assistant proposes dividing both sides of 3x2−12x=03x^2 - 12x = 0 by 3x3x to solve it. Critique this method, and interpret what the missing root represents in the context of the garden bed. [2]
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18QuestionUnderstanding Literal EquationsAssessment Practice
6 marks~9 minCriterion B
A ball is dropped from rest and falls freely under gravity. The distance fallen is recorded below.

Time (s): 0, 1, 2, 3, 4

Distance (m): 0, 4.9, 19.6, 44.1, 78.4
a
Analyse the data to identify the relationship between distance fallen and time. [2]
b
Deduce a formula for the distance dd (in metres) fallen after tt seconds, justifying your formula using values from the table. [2]
c
A sensor is placed 130 m below the drop point. Advise whether the ball has reached the sensor after 5 seconds, supporting your recommendation with a calculation. [2]

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19QuestionCommon Mistakes in RearrangingAssessment Practice
4 marks~6 minCriterion C
A mobile data plan charges a monthly fee according to the formula C=3d+2C = 3d + 2, where CC is the total cost in dollars and dd is the data used in gigabytes. A customer receives a bill of C=5C = 5 dollars and wants to find dd.

A classmate claims the formula can be rearranged to give d=3C+2d = 3C + 2.
a
Show that the correct rearrangement of C=3d+2C = 3d + 2 to express dd in terms of CC is d=C−23d = \dfrac{C - 2}{3}. [1]
b
Explain the error in the classmate's rearrangement d=3C+2d = 3C + 2. [1]
c
Using both formulas with C=5C = 5, calculate the two values of dd and advise the customer which result to trust, justifying your answer using the cost structure of the plan. [2]
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20QuestionRearranging to Make a Variable the SubjectAssessment Practice
4 marks~6 minCriterion D
The graph shows the linear relationship between degrees Celsius (CC) and degrees Fahrenheit (FF). The line passes through (0,32)(0, 32) and (100,212)(100, 212).
a
Deduce the equation of the line in the form F=mC+bF = mC + b. [1]
b
Rearrange your equation to make CC the subject. [1]
c
A storage facility for temperature-sensitive medication requires the internal temperature to remain below 25 °C. A sensor records F=77F = 77 °F. Advise the facility manager whether the storage requirement is currently being met, justifying your answer with a calculation. [2]
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21QuestionMulti-Step Problems with VariablesAssessment Practice
4 marks~6 minCriterion D
A solar panel installation company charges a connection fee plus a fixed rate per panel installed. The graph shows total cost yy (dollars) against number of panels xx, passing through (0, 50)(0,\ 50) and (10, 200)(10,\ 200).
a
Calculate the rate charged per panel. [1]
b
Deduce the equation of the line in the form y=mx+cy = mx + c. [1]
c
A community centre has a budget of 420 dollars for this installation. Advise the centre manager whether the budget is sufficient for 25 panels, justifying your answer with a calculation. [2]
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22QuestionSolving One-Step and Two-Step EquationsAssessment Practice
8 marks~12 minCriterion B
A mobile data plan charges a fixed monthly fee of bb dollars plus aa dollars per gigabyte used. Last month, a customer paid exactly 17 dollars in total.
a
With b=5b = 5, calculate the number of gigabytes xx used for each per-gigabyte rate: a=2,3,4,6,12a = 2, 3, 4, 6, 12. Show your algebraic working for each case. [2]
b
With a=3a = 3, calculate the number of gigabytes xx used for each fixed fee: b=2,8,11,14,17b = 2, 8, 11, 14, 17. Show your algebraic working for each case. [2]
c
Analyse how changes in aa affect xx using your results from part (a), and how changes in bb affect xx using your results from part (b). [2]
d
The plan is advertised as suitable for customers who use at least 1 gigabyte per month. With b=5b = 5, advise whether a per-gigabyte rate of a=15a = 15 keeps the plan suitable for a customer who paid exactly 17 dollars. Justify your answer using the pattern identified in part (c). [2]

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23QuestionSolving One-Step and Two-Step EquationsAssessment Practice
6 marks~9 minCriterion D

A student tracks the battery percentage of a phone over time. The data collected is:

Time (h): 0, 1, 2, 3

Battery (%): 100, 82, 64, 46

The student models the battery drain with the equation B=100−18tB = 100 - 18t, where BB is the battery percentage and tt is the time in hours.

a
[2 marks] Use the model to predict the battery percentage after 4 hours.
b
[2 marks] Use the model to find the time when the battery reaches 0%.
c
[2 marks] Discuss two limitations of using this linear model to represent the phone's battery drain over time.
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24QuestionSolving One-Step and Two-Step EquationsAssessment Practice
4 marks~6 minCriterion C
A mobile data plan charges a fixed monthly fee of 4 dollars plus a cost per gigabyte used. In January, a customer used 6 GB and paid a total usage fee modelled by:

x+4=10x + 4 = 10

In February, the customer doubled their data usage, giving a total usage fee modelled by:

2x+4=162x + 4 = 16

where xx is the cost per gigabyte in dollars.
a
Calculate the value of xx for each equation. [2]
b
Explain why Equation B requires one more algebraic step than Equation A to isolate xx. [1]
c
Justify whether the February plan offers the customer better value per gigabyte than the January plan. [1]
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