Equations, Inequalities & Formulae — Free IB MYP2 Mathematics Questions

1QuestionSolving One-Step InequalitiesConcept Practice

2 marks~3 minCriterion B

Each inequality below has the same solution,

x < 4

.

x + 3 < 7 \quad x + 5 < 9 \quad x + 2 < 6 \quad x + 4 < 8

a

Identify the relationship between the number added to

x

and the boundary point 4 in each inequality. [1]

b

Using this relationship, calculate the solution to

x + 6 < 10

. [1]

Solutions

2QuestionGraphing Inequalities on a Number LineConcept Practice

3 marks~5 minCriterion A

A number line has an open circle at

-2

with shading extending to the right.

a

Identify the inequality shown by the number line. [1]

b

Explain what the open circle at

-2

tells you about the values that satisfy the inequality. [1]

c

Compare the solution sets of

x > -2

and

x \geq -2

. State one way they are the same and one way they differ. [1]

Solutions

3QuestionChecking Solutions in the Context of the ProblemConcept Practice

2 marks~3 minCriterion C

Tom has 50 dollars. He buys 3 notebooks, each costing

x

dollars, and receives 5 dollars in change. This situation is represented by:

3x + 5 = 50

A student claims each notebook costs

x = 15

dollars.

a

Calculate the value of the left-hand side of the equation when

x = 15

. [1]

b

Compare your result with the right-hand side and state whether the student's claim is correct. [1]

Solutions

4QuestionCreating and Solving Multi-Step Word ProblemsConcept Practice

3 marks~5 minCriterion A

A taxi charges a fixed starting fee plus a cost per kilometre. The graph shows total cost

C

(in dollars) against distance

d

(in kilometres). The line passes through

(0,\ 5)

and

(10,\ 25)

.

a

Identify the starting fee and the cost per kilometre. [1]

b

Write an equation for

C

in terms of

d

. [1]

c

A rival company charges

C = 3d + 2

. Compare the cost of each company for a 10 km trip, showing your working. [1]

Solutions

5QuestionCreating Formulae from Contextual SituationsConcept Practice

2 marks~3 minCriterion A

The diagram shows a rectangle with side lengths

x + 3

and

x

.

a

Write an expression for the perimeter of the rectangle in terms of

x

. [1]

b

Calculate the perimeter when

x = 4

. [1]

Solutions

6QuestionApplying the Speed Distance Time FormulaConcept Practice

2 marks~3 minCriterion A

A car travels from City P to City Q, a distance of 240 km, in 3 hours.

The speed formula is:

\text{speed} = \frac{\text{distance}}{\text{time}}

a

Identify the values of distance and time given in the question. [1]

b

Calculate the speed of the car in km/h. [1]

Solutions

7QuestionUnderstanding Equality and BalanceConcept Practice

2 marks~3 minCriterion A

A balance scale has a 5 kg weight on the left side and two weights of 2 kg and 3 kg on the right side.

a

State an equation, using an equals sign, that represents this balanced scale. [1]

b

Explain why the scale is balanced. [1]

Solutions

8QuestionSolving One-Step Addition and Subtraction EquationsConcept Practice

3 marks~5 minCriterion C

A part-time job pays a fixed amount per hour. The graph shows the relationship between hours worked (

h

) and total earnings (

E

) in dollars. The line passes through

(0, 0)

and

(5, 75)

.

a

Identify the rate of pay per hour. [1]

b

Write an equation for

E

in terms of

h

. [1]

c

A friend says: "Working 8 hours earns exactly 100 dollars at this job." Explain whether your friend is correct, using your equation. [1]

Solutions

9QuestionExpanding Brackets Before SolvingConcept Practice

2 marks~3 minCriterion B

Look at the four equations below.

2(x + 1) = 2x + 2

3(x + 2) = 3x + 6

4(x + 3) = 4x + 12

5(x + 4) = 5x + 20

a

Identify the pattern in how the left side expands to give the right side. [1]

b

State the general rule for expanding

a(x + b)

. [1]

Solutions

10QuestionSolving Equations Involving FractionsConcept Practice

3 marks~5 minCriterion A

The graph shows the line

y = \dfrac{1}{2}x + 3

.

a

Identify the

x

-coordinate of the point on the line where

y = 7

. [1]

b

Calculate the value of

x

by solving the equation

\dfrac{1}{2}x + 3 = 7

, showing all working. [1]

c

Compare your answer to part (b) with the value you read from the graph in part (a), and explain what this tells you about the two methods. [1]

Solutions

11QuestionGraphing Inequalities on a Number LineAssessment Practice

6 marks~9 minCriterion D

A city planner uses the inequality

x \leq 5

to regulate the height (in metres) of new buildings near an airport.

a

State what the closed circle at

x = 5

means on a number line. [1]

b

Draw the inequality

x \leq 5

on a number line scaled from 0 to 10. [2]

c

Identify one limitation of using a single inequality to control building heights, and explain how it could affect flight safety. [3]

Solutions

12QuestionUnderstanding Inequality SymbolsAssessment Practice

6 marks~9 minCriterion C

A student solved

-3 < 2x + 1 \leq 7

and drew a number line with an open circle at

-2

and a closed circle at

3

, but shaded the region outside the interval.

The student's algebraic steps were:

Step 1:

-3 < 2x + 1 \leq 7

Step 2:

-4 < 2x \leq 6

Step 3:

-2 < x \leq 3

a

Identify the error in the student's number line. [1]

b

Calculate the correct solution to

-3 < 2x + 1 \leq 7

, showing each algebraic step. [2]

c

Explain why each inequality symbol (

<

or

\leq

) stays the same throughout your working, and describe what the open and closed circles on your number line represent. [3]

Solutions

13QuestionChecking Solutions in the Context of the ProblemAssessment Practice

6 marks~9 minCriterion B

A party planner uses the formula

C = 7n + 8

to estimate the total cost in dollars

C

for

n

guests.

Number of guests ( $n$ )	1	2	3	4
Total cost in dollars ( $C$ )	15	22	29	36

a

Calculate

C

using

C = 7n + 8

for each value of

n

in the table. State whether the formula matches all four values. [2]

b

Using

C = 7n + 8

, calculate the total cost for 10 guests. Explain whether this result is reasonable for a real-world party. [2]

c

The planner later finds the actual costs are:

(1, 14)

,

(2, 21)

,

(3, 28)

,

(4, 35)

. Identify the correct formula and explain how you found it. [2]

Solutions

14QuestionTranslating Word Problems into EquationsAssessment Practice

3 marks~5 minCriterion D

A plumber charges a fixed call-out fee plus an hourly rate. The graph of total cost

C

(dollars) against hours worked

h

passes through

(0,\ 50)

and

(4,\ 170)

.

a

Identify the

y

-intercept and state what it represents in this context. [1]

b

Calculate the slope of the line and state what it represents in this context. [1]

c

Write an equation for

C

in terms of

h

. Compare the cost of this plumber with a second plumber who charges no call-out fee but 35 dollars per hour, for a 6-hour job. State which plumber is cheaper. [1]

Solutions

15QuestionSolving Problems Involving Real-Life FormulaeAssessment Practice

5 marks~8 minCriterion D

A car's fuel use (in litres) is modelled by the formula

F = d \times 0.08

where

d

is the distance driven in kilometres. The fuel tank holds 60 litres.

a

Calculate the fuel needed for a 350 km highway trip. [1]

b

Explain why the model may underestimate fuel use during stop-and-go city driving. Identify one assumption the model makes and describe how city driving breaks that assumption. [2]

c

A driver completes a 350 km trip: half on the highway and half in the city. Compare the fuel predicted by the original model with the fuel predicted by the improved model

F = (d_{\text{city}} \times 0.12) + (d_{\text{highway}} \times 0.08)

, and state which gives the higher estimate. [2]

Solutions

16QuestionSolving Problems Involving Real-Life FormulaeAssessment Practice

6 marks~9 minCriterion B

The perimeter of a rectangle is given by

P = 2(l + w)

, where

l

is the length and

w

is the width.

a

The length and width of a rectangle sum to 10 cm. Calculate the perimeter when

l = 3

cm and when

l = 7

cm. [2]

b

Describe what you notice about the perimeters in part (a). State a general rule linking the perimeter to any fixed sum

S

of length and width. [2]

c

A second rectangle has

l + w = 15

cm. Using your rule, compare the perimeter of this rectangle with that of a rectangle where

l + w = 10

cm. Explain which is larger and by how much. [2]

Solutions

17QuestionUsing Standard Formulae Area Volume SpeedAssessment Practice

6 marks~9 minCriterion C

A rectangular prism has a length of 12 cm, a width of 8 cm, and a height of 10 cm. A cylindrical hole (a circular tunnel) with a radius of 3 cm is drilled vertically through the prism from top to bottom.

Use

V = l \times w \times h

and

V = \pi r^2 h

, where

\pi \approx 3.14

.

a

State the volume of the rectangular prism. [1]

b

Calculate the volume of the cylindrical hole. [2]

c

A student says the remaining solid has a volume of about 700 cm³. Calculate the correct remaining volume, rounded to the nearest cm³, and explain whether the student is right or wrong. [3]

Solutions

18QuestionSolving Finance-Related Formula ProblemsAssessment Practice

5 marks~8 minCriterion C

A student claims: "If I deposit 500 dollars in a savings account at 4% annual compound interest (interest calculated on the growing total each year), after 3 years I will have exactly 562.43 dollars."

Using the formula

A = P(1 + r)^t

:

a

Identify the values of

P

,

r

, and

t

. [1]

b

Calculate the actual amount

A

after 3 years. Show all steps. [2]

c

Compare your answer with the student's claim and explain whether the claim is accurate. [2]

Solutions

19QuestionUsing the Formula for Area of ShapesAssessment Practice

5 marks~8 minCriterion B

A shear transforms a rectangle into a parallelogram by sliding the top side horizontally while keeping the base and height the same.

Use the three rectangles below to investigate the conjecture: a rectangle and a parallelogram with the same base and height have equal area.

Rectangle 1	base $= 5$ cm	height $= 3$ cm
Rectangle 2	base $= 7$ cm	height $= 4$ cm
Rectangle 3	base $= 6$ cm	height $= 5$ cm

a

Calculate the area of each rectangle. Record your results as labelled rows with the headings: Rectangle, Base (cm), Height (cm), Area (cm

^2

). [2]

b

Each rectangle is sheared into a parallelogram with the same base and height. Calculate the area of each parallelogram using

A = \text{base} \times \text{height}

, and add an "Area of parallelogram (cm

^2

)" row to your table. [1]

c

Compare the two sets of areas in your table. State the pattern you observe and explain why the formula gives the same result for both shapes. [2]

Solutions

20QuestionApplying the Speed Distance Time FormulaAssessment Practice

4 marks~6 minCriterion D

A car travels through a city for 30 minutes at a constant speed of 40 km/h. Using the formula

\text{distance} = \text{speed} \times \text{time}

a student calculates the car travels 20 km.

a

Identify one reason why the actual distance travelled may differ from 20 km. [1]

b

Explain how using average speed instead of constant speed would improve this estimate. [2]

c

A second city journey lasts 45 minutes at an average speed of 30 km/h. Compare the two journeys in terms of distance travelled. [1]

Solutions

21QuestionSolving One-Step Addition and Subtraction EquationsAssessment Practice

6 marks~9 minCriterion B

Solve each equation by using the inverse operation — the opposite operation that "undoes" the one already in the equation.

x + 5 = 12 \qquad y - 3 = 9 \qquad z + 7 = 15 \qquad w - 4 = 11

a

Identify the inverse operation used to solve each equation and state the number applied to both sides. [2]

b

Describe the relationship between the operation in the original equation and the inverse operation used to solve it. [2]

c

Compare the two equation forms

x + a = b

and

x - a = b

. State the general rule for solving each form. [2]

Solutions

22QuestionSolving One-Step Multiplication and Division EquationsAssessment Practice

6 marks~9 minCriterion D

A farmer has 350 kg of fertilizer. Each bag holds 4.8 kg, and 5% of the fertilizer is lost during spreading (meaning only 95% is usable).

a

Solve

5x = 350

. State the value of

x

. [1]

b

Calculate the actual number of bags needed. Use the usable amount of fertilizer and the real bag weight of 4.8 kg. Show your working. [3]

c

Identify two assumptions made by the equation

5x = 350

and explain how each assumption affects the answer. [2]

Solutions

23QuestionClearing Fractions in EquationsAssessment Practice

3 marks~5 minCriterion C

The graph shows the lines

y = \dfrac{x+1}{3}

and

y = \dfrac{2x-1}{2}

.

a

Identify the x-coordinate of the point where the two lines intersect. [1]

b

Explain how this x-coordinate gives the solution to the equation

\dfrac{x+1}{3} = \dfrac{2x-1}{2}

. [1]

c

Calculate the exact solution algebraically and compare it with your answer in part (a). [1]

Solutions

24QuestionSolving Equations with Variables on Both SidesAssessment Practice

2 marks~3 minCriterion D

A student compares two mobile phone plans:

Plan A: 20 dollars per month + 0.10 dollars per text
Plan B: 10 dollars per month + 0.15 dollars per text

The equation

20 + 0.10x = 10 + 0.15x

shows when the plans cost the same, giving a solution of

x = 200

texts.

a

Identify one real-world limitation of this model. [1]

b

Explain how this limitation could affect the break-even result of 200 texts. [1]

Solutions

Equations, Inequalities & Formulae — Free MYP2 Mathematics Practice Questions