Inequalities

Bernie is purchasing $x$ packets of seeds and $y$ plants for his garden. He intends to buy a greater number of seed packets than plants, so the inequality $x > y$ represents this. He also wants fewer than 10 packets of seeds and no fewer than 2 plants.

Feb/March 2017

Klaus purchases $x$ silver balloons and $y$ gold balloons for a party. He gets more gold balloons than silver balloons, at least 15 silver balloons, fewer than 50 gold balloons, and in total no more than 70 balloons.

Feb/March 2018

State the values of $n$ that satisfy $7 < n \leq 15$.

Feb/March 2025

The number line extends from -6 to 6. An unfilled circle is marked at $x = -3$ and a solid circle is marked at $x = 4$, with a solid segment joining them.

Feb/March 2025

A graph is shown with the $x$-axis horizontal and the $y$-axis vertical. Two straight lines cross, and a dashed horizontal line is included. The shaded area lies outside the central unshaded area.

May/June 2015

Solve this inequality: $n + 7 < 5n - 8$.

May/June 2016

Solve the inequality $\frac{x}{3} + 5 > 2$.

May/June 2016

Solve this inequality: $3n - 11 > 5n - 18$.

May/June 2017

The diagram presents a coordinate grid with the axes marked $x$ and $y$, together with a number of straight lines.

May/June 2017

Solve for $x$ in the inequality $x + 13 > 3x + 7$.

May/June 2017

Solve the inequality $3n - 5 > 17 + 8n$.

May/June 2018

The diagram displays the inequalities $y \leq -\frac{1}{2}x + 6$, $y \geq 3x - 4$, $x + y \geq 5$.

May/June 2019

Using shading to remove the regions that are not wanted, draw and label region $R$ so that it satisfies these three inequalities: $y \leq 2$, $x < 3$, $y \leq x + 4$. The grid is drawn with labelled $x$- and $y$-axes.

May/June 2019

The integer $x$ satisfies $-3 \leq 2x - 1 < 3$.

May/June 2021

Solve the inequality $12x - 3 \ge 4x + 13$.

May/June 2022

Find every positive integer that satisfies the inequality $3n - 8 > 5n - 15$.

May/June 2022

$x$ is an integer satisfying $x \geq -3$ and $x < 3$.

May/June 2023

Finish this sentence about the value of $m$.

May/June 2023

A tailor produces $x$ dresses and $y$ shirts during one week. Over that week, he produces at least 4 dresses, no more than 7 shirts, fewer than 14 dresses and shirts in total, and the number of shirts he produces is greater than $\frac{2}{3}$ of the number of dresses. One inequality that represents this information is $x \geq 4$.

May/June 2023

State the inequalities that define the unshaded region, $R$.

May/June 2024

A baker decorates $x$ small cakes and $y$ large cakes. Over the course of one day, he decorates: • at most 16 small cakes • fewer than 10 large cakes • more small cakes than large cakes • no more than 24 cakes altogether. One inequality that represents this information is $x \leq 16$.

May/June 2024

State the value of $P$ that satisfies the inequality $13 < P < 19$.

May/June 2025

A number line has a filled dot at $-4$ and a hollow circle at $3$, with the section between them shaded.

May/June 2025

Represent the inequality $-3 < x \leq 2$ on the number line.

May/June 2025

Shown on the number line is a filled circle at $-1$ and a hollow circle at $4$, with a line drawn between them in the direction of increasing $y$.

May/June 2025

The coordinate diagram displays an unshaded triangular region $R$. Its boundaries are the $y$-axis, the vertical line $x=1$, the $x$-axis, and a straight line descending from $(0,4)$ to $(4,0)$. The shaded parts are outside $R$.

Oct/Nov 2016

Determine the positive integers that make the inequality $t+2>3t-6$ true.

Oct/Nov 2016

A shaded coordinate grid is shown, with region R left unshaded. A solid vertical boundary appears at $x = 2$. A solid oblique line slopes down from $(0,6)$ to $(10,1)$. A dashed line slopes up from $(0,0)$ to $(7,7)$. The axes are labelled $x$ and $y$.

Oct/Nov 2016

Across one week, Neha spends $x$ hours cooking and $y$ hours cleaning. The time she spends cleaning is at least as much as the time she spends cooking. This may be written as $y \ge x$. In total, she spends no more than 16 hours cooking and cleaning. She spends at least 4 hours cooking.

Oct/Nov 2017

Find the integers that satisfy the inequality $-5 < 2n - 1 \leq 5$.

Oct/Nov 2017

Solve the inequality $7 - 8x \geq 19 + 2x$.

Oct/Nov 2017

Find the integer values of n for which the inequality $15 \leq 4n < 28$ is true.

Oct/Nov 2018

A coordinate plane is displayed with the $x$-axis running from $0$ to $4$ and the $y$-axis running from $0$ to $8$. The lines drawn are $y = 2x + 1$, $y = 4 - x$, together with the vertical lines $x = 2$ and $x = 3$. The axes are marked $x$ and $y$.

Oct/Nov 2018

Solve the inequality $7m - 2 \ge 19$.

Oct/Nov 2018

A coordinate grid is displayed, together with a straight line labelled $2x + y = 6$.

Oct/Nov 2019

Solve this inequality: $\frac{x}{2} - 13 > 12 + 3x$.

Oct/Nov 2019

A car hire firm owns $x$ small cars and $y$ large cars. Altogether, the firm has at least 6 cars. The number of large cars is no more than the number of small cars. The greatest possible number of small cars is 8.

Oct/Nov 2019

Solve for $x$: $4 - 3x \ge \frac{6 - x}{5}$.

Oct/Nov 2021

$x$ is an integer subject to $x > -4$ and $x \leq 2$.

Oct/Nov 2022

State the inequality represented by the number line.

Oct/Nov 2022

Find the value of $P$ for $k=3$.

Oct/Nov 2022

Solve the inequality $4(2x-3) \ge 43 + 3x$.

Oct/Nov 2023

A coordinate grid is displayed, with the x-axis running from $-5$ to $7$ and the y-axis running from $-3$ to $8$.

Oct/Nov 2023

The graph of $y = 2x + 1$ is shown on the grid. By shading the parts of the grid that are not needed, identify and mark the region $R$ that obeys these inequalities: $y \ge 2x + 1$ $y \ge 1$ $4x + 3y < 12$

Oct/Nov 2023

A coordinate grid is displayed, with the $x$-axis and $y$-axis labelled from $-5$ to $5$. Region $R$ is defined by the inequalities $-3 < y \le 2$ and $y \le x - 1$.

Oct/Nov 2024

A firm produces scientific calculators and graphic calculators. On each day, it manufactures $x$ scientific calculators and $y$ graphic calculators. The following inequalities represent the daily numbers produced: $x < 180$, $y \leq 90$, $x + y \leq 240$.

Oct/Nov 2024

A number line has a filled circle at -5 and an unfilled circle at 1, with the stretch between those points shaded.

Oct/Nov 2025

The graph encloses region $R$ with boundaries at $y=4$, $y=3$, $x=2.5$ and $y=2x$.

Oct/Nov 2025

Represent $-4 < x \leq 3$ on the number line.

Oct/Nov 2025

Show the inequality $x \geq 3$ on the number line.

Oct/Nov 2025

Write down the inequality shown by the diagram.

Oct/Nov 2025