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Equations, Inequalities and Formulae

Equations, Inequalities and Formulae — Free MYP5 Mathematics (Standard) Practice Questions

1QuestionSolving Linear Inequalities in One VariableConcept Practice
4 marks~6 minCriterion C
A city regulation states that a food truck must be located more than 4 metres from the nearest fire hydrant. A number line is provided showing all permitted distances xx (in metres) from a hydrant.
a
Interpret the number line: explain what the open circle at x=4x = 4 and the arrow pointing right each indicate about the solution set. [1]
b
Write an inequality in the form ax+b>cax + b > c, where aa, bb, and cc are integers and a≠1a \neq 1, that represents the same solution set. [1]
c
A rival truck parks at x=4.0x = 4.0 m. Justify whether this truck complies with the regulation. [2]
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2QuestionSolving Linear Inequalities in One VariableConcept Practice
2 marks~3 minCriterion A
A drone delivery service guarantees same-day dispatch only when the daily order queue contains fewer than 3 hours of processing time remaining.

The inequality x<3x < 3 models this condition, where xx represents hours of processing time remaining.
a
Represent the inequality x<3x < 3 on a number line. [1]
b
The current processing queue stands at exactly 3 hours. Justify whether the drone service will guarantee same-day dispatch today. [1]
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3QuestionSolving by Elimination MethodConcept Practice
4 marks~6 minCriterion A
A city planner models two pedestrian pathways on a coordinate grid. Pathway P follows the line 2x+y=82x + y = 8 and Pathway Q follows the line x−y=1x - y = 1. The pathways intersect at a single point, which marks the location of a proposed information kiosk.
a
Use the elimination method to find the exact coordinates of the intersection point. [2]
b
The kiosk must be placed at least 2.5 units from the origin to avoid underground utilities. Justify whether the proposed location satisfies this requirement. [2]
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4QuestionSolving by FactorizationConcept Practice
2 marks~3 minCriterion A
A landscape architect is designing a rectangular garden with a length of (x+4)(x + 4) m and a width of (x+2)(x + 2) m. The total area of the garden must be exactly 35 m235 \text{ m}^2.
a
Show that the area condition gives the equation x2+6x−27=0x^2 + 6x - 27 = 0. [1]
b
Solve x2+6x−27=0x^2 + 6x - 27 = 0 by factorization. Hence, deduce the actual dimensions of the garden and justify which solution is valid in this context. [1]
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5QuestionInterpreting 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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6QuestionMulti-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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7QuestionEquations Involving Brackets and FractionsConcept Practice
2 marks~3 minCriterion B
A mobile data plan charges a fixed monthly fee. The total cost CC (in dollars) after nn months satisfies the sequence of equations below, where xx is the monthly fee.

x2=5,x+13=5,x+24=5,x+35=5\frac{x}{2} = 5, \quad \frac{x+1}{3} = 5, \quad \frac{x+2}{4} = 5, \quad \frac{x+3}{5} = 5

Analyse the pattern in the sequence. Write the next equation and deduce the value of xx it gives. [2]

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8QuestionEquations Involving Brackets and FractionsConcept Practice
2 marks~3 minCriterion A
A personal trainer designs a 45-minute session split into three identical blocks. Each block includes a 5-minute warm-up plus 2x2x minutes of circuit training, where xx is the number of circuit rounds.
a
Show that the total session time gives the equation 3(2x+5)=453(2x + 5) = 45, and solve for xx. [1]
b
The trainer states: "Each block has time for at least 4 circuit rounds." Using your value of xx, justify whether this statement is correct. [1]
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9QuestionGraphical 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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10QuestionGraphical Representation of InequalitiesAssessment Practice
4 marks~6 minCriterion D

A farmer has 100 hectares of land and a budget of 120000 dollars to plant two crops: wheat and corn. Planting one hectare of wheat costs 1000 dollars, while one hectare of corn costs 1500 dollars. The profit per hectare of wheat is 5000 dollars, and the profit per hectare of corn is 8000 dollars.

a
Let xx be the number of hectares of wheat and yy be the number of hectares of corn. Write down two inequalities that represent the constraints on the land and budget.
b
Sketch a graph showing the feasible region defined by these inequalities, with xx on the horizontal axis and yy on the vertical axis. Label the axes clearly.
c
The farmer wants to maximize their profit. Discuss the strengths and limitations of using this linear programming model to determine the optimal number of hectares of each crop to plant. Consider factors such as unpredictable weather patterns, fluctuating market prices, and the assumption of constant profit per hectare.
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11QuestionSolving by Substitution MethodAssessment Practice
4 marks~6 minCriterion B
A logistics company tracks two delivery routes. Each day, the total distance covered by both routes is aa km, and the difference between the longer and shorter route is bb km. The distances xx (longer) and yy (shorter), in km, satisfy:
x+y=ax−y=bx + y = a \qquad x - y = b
a
Deduce the values of xx and yy for each system. [1]

System 1: a=5, b=1a = 5,\ b = 1 \quad System 2: a=7, b=3a = 7,\ b = 3

System 3: a=10, b=4a = 10,\ b = 4 \quad System 4: a=12, b=2a = 12,\ b = 2
b
Deduce general formulas for xx and yy in terms of aa and bb. [1]
c
Prove that your formulas satisfy both original equations. [1]
d
The company requires the shorter route to be at least 30% of the total daily distance. For a=40a = 40 and b=10b = 10, advise the company whether this scheduling requirement is met, justifying your answer using your formulas. [1]

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12QuestionSolving GraphicallyAssessment Practice
4 marks~6 minCriterion C
A city planner models two road-cost proposals using the equations y=2x−1y = 2x - 1 and y=−x+5y = -x + 5, where xx is years after the project starts and yy is cost in millions of dollars. The graph of both lines is shown.
a
State the coordinates of the intersection point of the two lines. [1]
b
Interpret what the intersection point represents in terms of the two cost models. [1]
c
The planner states: "The two proposals reach equal cost at exactly year 2." Justify whether this claim is correct, showing your algebraic reasoning. [2]
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13QuestionSolving by Elimination MethodAssessment Practice
4 marks~6 minCriterion D
A small business owner models monthly revenue and cost (in dollars) as:

R=15xC=2000+8xR = 15x \qquad C = 2000 + 8x

where xx is the number of units sold, the selling price is 15 dollars per unit, the fixed monthly rent is 2000 dollars, and the variable material cost is 8 dollars per unit.
a
Use the elimination method to find the break-even value of xx, where R=CR = C. State your answer as a whole number of units and justify your rounding. [2]
b
The business introduces a bulk discount for large orders and experiences seasonal demand fluctuations. Advise the business owner whether this linear model should be used to predict profit under these conditions, citing the specific assumptions that are affected. [2]
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14QuestionSolving by FactorizationAssessment Practice
2 marks~3 minCriterion B
Examine the following factorized quadratics:

x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x+2)(x+3)
x2+7x+12=(x+3)(x+4)x^2 + 7x + 12 = (x+3)(x+4)
x2+9x+20=(x+4)(x+5)x^2 + 9x + 20 = (x+4)(x+5)
x2+11x+30=(x+5)(x+6)x^2 + 11x + 30 = (x+5)(x+6)

Analyse the relationship between the constant term, the coefficient of xx, and the integers in the brackets across all four expressions.

Deduce the factorized form of x2+13x+42x^2 + 13x + 42, clearly stating your conjecture in written form. [2]

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15QuestionSolving 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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16QuestionSolving by FactorizationAssessment Practice
2 marks~3 minCriterion D
In a soccer match, a player kicks a ball from ground level. Its height, hh metres, after tt seconds is modelled by

h=−5t2+20t.h = -5t^2 + 20t.
a
Factorise −5t2+20t-5t^2 + 20t and hence deduce the time at which the ball returns to the ground. [1]
b
The referee rules the ball "out of play" if it lands beyond t=5t = 5 seconds. Using your result from part (a), advise the referee whether the ball should be ruled out of play, and state one limitation of the quadratic model that could affect this judgement in a real match. [1]
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17QuestionApplications in Geometry and Physics FormulasAssessment Practice
4 marks~6 minCriterion B
A construction team is checking whether a triangular brace is truly right-angled before installing it. They measure the three sides as 2.5 m, 6 m, and 6.5 m.

Before applying this to the brace, investigate the relationship between side lengths in right triangles using the four Pythagorean triples below.

Triangle 1: (3,4,5)(3, 4, 5) — Triangle 2: (5,12,13)(5, 12, 13) — Triangle 3: (6,8,10)(6, 8, 10) — Triangle 4: (7,24,25)(7, 24, 25)
a
For each triangle, calculate the sum of the squares of the two shorter sides and the square of the longest side. [1]
b
Deduce a general equation relating the three side lengths of any right triangle. [1]
c
Using your equation from (b), advise the construction team whether the brace with sides 2.5 m, 6 m, and 6.5 m should be approved for installation, justifying your answer with full working. [2]

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18QuestionRearranging with Fractions and BracketsAssessment Practice
2 marks~3 minCriterion A
A surveyor models a triangular plot of land. The three sides measure a=5a = 5 m, b=7b = 7 m, and c=6c = 6 m, where angle θ\theta is opposite side cc. The cosine rule states:

c2=a2+b2−2abcos⁡θc^2 = a^2 + b^2 - 2ab\cos\theta
a
Rearrange the cosine rule to make cos⁡θ\cos\theta the subject. [1]
b
Calculate θ\theta and hence justify whether the triangular plot contains any obtuse angle. [1]
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19QuestionApplications in Geometry and Physics FormulasAssessment Practice
4 marks~6 minCriterion D
A photographer uses a thin lens with a focal length of f=50f = 50 mm. The lens formula

1f=1u+1v\frac{1}{f} = \frac{1}{u} + \frac{1}{v}

relates object distance uu and image distance vv.
a
Show that v=ufu−fv = \dfrac{uf}{u - f}. [1]
b
An object is placed at u=200u = 200 mm from the lens. Calculate the image distance vv. [1]
c
A sharp image requires vv to fall within 2 mm of the calculated value. Assess whether the thin-lens model is reliable enough to guarantee this precision for a professional camera lens, justifying your answer with two distinct physical limitations of the model. [2]
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20QuestionCommon Mistakes in RearrangingAssessment Practice
2 marks~3 minCriterion C
A landscape architect is designing a triangular garden bed. The area must be exactly 30 m230\ \text{m}^2 and the base measures 6 m6\ \text{m}.

To find the required height hh, a student rearranges A=12bhA = \frac{1}{2}bh and writes:

h=12×30×6h = \frac{1}{2} \times 30 \times 6
a
Identify the error in the student's rearrangement. [1]
b
Deduce the correct value of hh. [1]
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21QuestionAge Distance and Geometry-Based ProblemsAssessment Practice
5 marks~8 minCriterion A
A rectangular park has length (3x+5)(3x + 5) m and width (2x−3)(2x - 3) m, where xx represents the age in years of a student. The perimeter of the park is 124 m. A straight diagonal path connects two opposite corners of the park.
a
Show that the perimeter of the park gives the equation 10x+4=12410x + 4 = 124, and hence deduce the value of xx. [2]
b
Using your value of xx, calculate the length and width of the park. [1]
c
The park's management states that any diagonal path longer than 45 m requires safety lighting. Using Pythagoras' theorem, calculate the length of the diagonal path correct to 2 decimal places, and hence justify whether safety lighting is required. [2]
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22QuestionChoosing Appropriate MethodsAssessment Practice
2 marks~3 minCriterion D
A small business pays a monthly electricity bill structured as follows: a fixed base charge of 50 dollars, a rate of 0.10 dollars per kWh for the first 500 kWh, and 0.15 dollars per kWh for any usage beyond 500 kWh.

A single linear equation C=50+0.10xC = 50 + 0.10x has been proposed to model the total monthly cost CC (in dollars) for xx kWh consumed.

Calculate the cost predicted by the linear model and the actual cost when x=800x = 800 kWh, then justify whether the single linear equation is a suitable model for this billing structure. [2]
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23QuestionWord Problems Involving Linear EquationsAssessment Practice
4 marks~6 minCriterion D
An architect is designing a rectangular garden bed with a perimeter of 94 cm. The length is 5 cm more than twice the width.
a
Write a linear equation in one variable that represents the perimeter condition of the rectangle. [1]
b
Determine the length and width of the rectangle. Show your working. [2]
c
The architect considers redesigning the garden bed as a square with the same perimeter. Advise the architect whether switching to the square design is worthwhile for maximising planting area. [1]
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24QuestionSolving One-Step and Two-Step EquationsAssessment Practice
4 marks~6 minCriterion C
A student solves 5x−10=205x - 10 = 20 and produces the following working:

Step 1: 5x=20−10=105x = 20 - 10 = 10

Step 2: x=105=2x = \dfrac{10}{5} = 2
a
Identify the step in which the error occurs and state what the student did incorrectly. [1]
b
Show that the correct solution is x=6x = 6. [2]
c
An engineer uses the equation 5x−10=205x - 10 = 20 to model the load (in kilonewtons) on a structural beam, where xx represents the number of support columns. Advise the engineer whether the student's solution should be used to determine the number of columns, justifying your answer in terms of structural safety. [1]
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