Equations and Inequalities

What an inequality is and how it differs from an equation, how to check whether a number is a solution, how to list whole-number solutions, and how to write an inequality from words.

Before you start

From Secondary 1 you know the four signs: $<$ is "is less than", $>$ is "is greater than", $\le$ is "is less than or equal to" and $\ge$ is "is greater than or equal to".

On a number line, the number further to the right is the greater. A statement with $\le$ is true when either "less than" or "equal to" is true, and $\ge$ likewise: $3 \le 3$ is true, but $3 < 3$ is false.

The open end of $<$ or $>$ faces the greater number, so $7 > 4$ and $4 < 7$ say the same thing.

Example 1: Comparing numbers

(a) Write $<$ , $>$ or $=$ between the two numbers of each pair.
(i) $-\frac{1}{2}$ and $-\frac{1}{3}$ (ii) $0.3$ and $\frac{1}{3}$ (iii) $-2.05$ and $-2.5$ (iv) $\frac{3}{4}$ and $0.75$
(b) Is each statement true or false?
(i) $-4 \ge -4$ (ii) $5 \le 3$ (iii) $-2 > -7$ (iv) $0 < -1$

Solution

(a)(i) As decimals, $-\frac{1}{2} = -0.5$ and $-\frac{1}{3} = -0.333\ldots$ Both are negative, and $-0.5$ is further from zero, so it is the smaller.

(ii) $\frac{1}{3} = 0.333\ldots$ , which is greater than $0.3$ .

(iii) Both are negative, and $-2.5$ is further from zero, so $-2.05$ is the greater.

(iv) $\frac{3}{4} = 0.75$ , so the two numbers are equal.

(b)(i) The two numbers are equal, and $\ge$ allows "equal to". (ii) 5 is greater than 3: it is neither less than 3 nor equal to it. (iii) $-2$ is to the right of $-7$ on the number line. (iv) 0 is greater than every negative number.

(a)(i) $-\frac{1}{2} < -\frac{1}{3}$ (ii) $0.3 < \frac{1}{3}$ (iii) $-2.05 > -2.5$ (iv) $\frac{3}{4} = 0.75$ (b)(i) True (ii) False (iii) True (iv) False

What an Inequality Is

An equation is a statement that two expressions are equal, such as $x + 5 = 8$ . An inequality is a statement that compares two expressions with one of the signs $<$ , $>$ , $\le$ or $\ge$ , such as $x + 5 > 8$ .

Both can contain an unknown. The equation $x + 5 = 8$ is true for one value of $x$ only, 3, and false for every other value. The inequality $x + 5 > 8$ is true for many values: for $x = 4$ , because $9 > 8$ ; for $x = 3.5$ , because $8.5 > 8$ ; for $x = 100$ ; and for every other number greater than 3.

Inequalities

An inequality states that one quantity is less than, greater than, less than or equal to, or greater than or equal to another. It is written with one of the signs $<$ , $>$ , $\le$ or $\ge$ .

The signs $<$ and $>$ do not allow the two sides to be equal. The signs $\le$ and $\ge$ do.

$a < b$ and $b > a$ say the same thing, and so do $a \le b$ and $b \ge a$ .

An inequality is read from left to right. $x \ge 5$ is read " $x$ is greater than or equal to 5". $3 < x$ is read "3 is less than $x$ ", which says the same as " $x$ is greater than 3".

It is usual to write the unknown on the left. To rewrite $3 < x$ that way, swap the two sides and turn the sign round with them: $x > 3$ . The open end of the sign still faces $x$ .

Example 2: Reading an inequality and turning it round

Write each inequality in words. Then write it with the unknown on the left.
(a) $7 \ge y$
(b) $-2 < p$
(c) $0 \le k$

Solution

(a) "7 is greater than or equal to $y$ ". So $y$ is less than or equal to 7: swapping the two sides turns $\ge$ into $\le$ .

(b) " $-2$ is less than $p$ ". So $p$ is greater than $-2$ .

(a) $y \le 7$ (b) $p > -2$ (c) $k \ge 0$

Equation or Inequality: One Solution or Many

A value of the unknown that makes an equation or an inequality true is a solution of it, and we say that it satisfies the equation or inequality. To solve an equation or an inequality is to find all its solutions.

The linear equation $x + 5 = 8$ has exactly one solution, $x = 3$ . The inequality $x + 5 > 8$ has infinitely many: every number greater than 3, whole or not. All of them together are written $x > 3$ , and this is the solution of the inequality.

The number 3 itself is not a solution of $x + 5 > 8$ , because $3 + 5 = 8$ and $8 > 8$ is false. But it is a solution of $x + 5 \ge 8$ , because $8 \ge 8$ is true. So the solution of $x + 5 \ge 8$ is $x \ge 3$ .

	Linear equation	Inequality
Sign	$=$	$<$ , $>$ , $\le$ or $\ge$
Example	$x + 5 = 8$	$x + 5 > 8$
Its solution	$x = 3$	$x > 3$
Number of solutions	exactly one	infinitely many

An expression such as $3y + 4$ has no sign at all. It is neither an equation nor an inequality: it has a value for each value of $y$ , but it states nothing that could be true or false.

Example 3: Equation, inequality or neither

Say whether each is an equation, an inequality or neither. For each equation and each inequality, say whether it has one solution or many.
(a) $2x - 1 = 9$
(b) $2x - 1 < 9$
(c) $3y + 4$
(d) $-4 \le k$
(e) $\frac{n}{2} = 6$

Solution

(a) It has the sign $=$ , so it is an equation. Only $x = 5$ makes it true: $2(5) - 1 = 9$ .

(b) It has the sign $<$ , so it is an inequality. $x = 0$ , $x = 1$ and $x = 4.5$ all satisfy it: for example, $2(0) - 1 = -1$ , and $-1 < 9$ . It has many solutions.

(d) It has the sign $\le$ , so it is an inequality. It says $k \ge -4$ , and $-4$ , 0 and 10 are three of its many solutions.

(e) It has the sign $=$ , so it is an equation. Only $n = 12$ makes it true.

(a) An equation, with one solution (b) An inequality, with many solutions (c) Neither: it is an expression (d) An inequality, with many solutions (e) An equation, with one solution

Is a Value a Solution?

To decide whether a number is a solution, put it in place of the unknown and work out each side. What is left is a statement about two numbers, and it is either true or false.

Checking a value

Step 1: Substitute the value for the unknown.

Step 2: Work out the left-hand side and the right-hand side.

Step 3: Decide whether the inequality between the two numbers is true. If it is, the value is a solution. If it is not, the value is not a solution.

Example 4: Substituting and checking

(a) Is $x = 3$ a solution of $2x + 1 > 7$ ?
(b) Is $x = 4$ a solution of $2x + 1 > 7$ ?
(c) Is $x = 3$ a solution of $2x + 1 \ge 7$ ?

Solution

(a) When $x = 3$ , the left-hand side is $2(3) + 1 = 7$ and the right-hand side is 7. $7 > 7$ is false, so $x = 3$ is not a solution.

(b) When $x = 4$ , the left-hand side is $2(4) + 1 = 9$ . $9 > 7$ is true, so $x = 4$ is a solution.

(c) When $x = 3$ , the left-hand side is 7 again. $7 \ge 7$ is true, because $\ge$ allows "equal to". So $x = 3$ is a solution of this inequality.

(a) No (b) Yes (c) Yes. Only the sign is different between (a) and (c): $x = 3$ makes the two sides equal, and only $\ge$ allows that.

Example 5: Which numbers satisfy an inequality?

Of the numbers $-2$ , 0, 1.5, 2 and 4, which satisfy
(a) $x < 2$ ,
(b) $3 - 2x \ge -1$ ?

Solution

(a) The numbers less than 2 are $-2$ , 0 and 1.5. 2 is not less than itself, and 4 is greater than 2.

(b) Substitute each number in turn and work out the left-hand side.

$x = -2$ : $3 - 2(-2) = 3 + 4 = 7\text{,}$ and $7 \ge -1$ is true.

$x = 0$ : $3 - 2(0) = 3$ , and $3 \ge -1$ is true.

$x = 1.5$ : $3 - 2(1.5) = 3 - 3 = 0\text{,}$ and $0 \ge -1$ is true.

$x = 2$ : $3 - 2(2) = 3 - 4 = -1\text{,}$ and $-1 \ge -1$ is true.

$x = 4$ : $3 - 2(4) = 3 - 8 = -5\text{,}$ and $-5 \ge -1$ is false, because $-5$ is to the left of $-1$ .

(a) $-2$ , 0 and 1.5 (b) $-2$ , 0, 1.5 and 2

Solutions That Must Be Whole Numbers

When the unknown counts something, such as people, tickets or bus trips, only whole numbers make sense. Then the solutions can be listed one by one.

The whole numbers are 0, 1, 2, 3, … The positive integers are 1, 2, 3, … So 0 is a whole number, but it is not a positive integer.

Example 6: Listing the solutions

(a) $x$ is a positive integer and $x < 4$ . List the possible values of $x$ .
(b) $x$ is a whole number and $x \le 4$ . List the possible values of $x$ .
(c) $n$ is a positive integer and $3n < 14$ . List the possible values of $n$ .

Solution

(a) The positive integers start at 1. They must be less than 4, and 4 itself is not allowed.

(b) The whole numbers start at 0, and $\le$ allows 4 itself.

(c) Try $n = 1, 2, 3, \ldots$ in turn: $3(1) = 3$ , $3(2) = 6$ , $3(3) = 9$ and $3(4) = 12$ are all less than 14, but $3(5) = 15$ is not. A larger $n$ gives a larger $3n$ , so every $n$ after 5 fails too.

(a) 1, 2, 3 (b) 0, 1, 2, 3, 4 (c) 1, 2, 3, 4

Writing an Inequality from Words

To write an inequality from a sentence, first choose a letter and say what it stands for, with its unit: "Let $m$ kg be the total mass." The letter stands for a number, and the unit is said once, in that sentence.

Choosing the sign

Step 1: Decide whether the words say greater or less.

Step 2: Decide whether the number named is itself allowed. If it is, use $\ge$ or $\le$ . If it is not, use $>$ or $<$ .

The words	Is 10 itself allowed?	Inequality
more than 10, greater than 10, above 10, exceeds 10	No	$x > 10$
less than 10, fewer than 10, below 10	No	$x < 10$
at least 10, not less than 10, no fewer than 10, a minimum of 10, 10 or more	Yes	$x \ge 10$
at most 10, not more than 10, up to 10, must not exceed 10, a maximum of 10, 10 or less	Yes	$x \le 10$

Example 7: Inequalities in everyday settings

Write an inequality for each sentence. Say what your letter stands for, with its unit.
(a) A lift can carry a load of at most 1000 kg.
(b) A film rated NC16 may be watched only by people aged 16 and above.
(c) To pass a test, a student's mark must be not less than 50.
(d) The temperature inside a fridge must stay below 5 °C.
(e) Ravi's bus fares for a week must not exceed $15.

Solution

(a) Let $m$ kg be the total mass of the people and things in the lift. "At most 1000" allows 1000 and anything less.

(b) Let $a$ years be the age of a person who may watch the film. "16 and above" allows 16 itself.

(d) Let $t$ °C be the temperature inside the fridge. "Below 5" does not allow 5 itself.

(e) Let Ravi's bus fares for the week be $f$ dollars. "Must not exceed 15" allows 15 and anything less.

(a) $m \le 1000$ (b) $a \ge 16$ (c) $s \ge 50$ (d) $t < 5$ (e) $f \le 15$

The words may describe an expression in the unknown, not the unknown itself. Write the expression first, then choose the sign.

Example 8: An inequality with an expression

(a) When 5 is subtracted from twice a number $x$ , the result exceeds 11. Write an inequality in $x$ .
(b) A school hall has 24 rows of chairs, with $r$ chairs in each row. It must seat no fewer than 600 students. Write an inequality in $r$ .
(c) Is $r = 25$ a solution of your inequality in (b)? Is $r = 24$ ?

Solution

(a) Twice the number is $2x$ , and subtracting 5 gives $2x - 5$ . "Exceeds 11" means "is greater than 11", so 11 itself is not allowed.

(b) The hall seats $24r$ students. "No fewer than 600" means 600 or more.

(c) When $r = 25$ : $24(25) = 600$ , and $600 \ge 600$ is true. When $r = 24$ : $24(24) = 576$ , and $576 \ge 600$ is false.

(a) $2x - 5 > 11$ (b) $24r \ge 600$ (c) $r = 25$ is a solution: 25 chairs in a row seat exactly 600 students. $r = 24$ is not: 24 chairs in a row seat only 576 students.

Common Mistakes

Reading $3 < x$ as " $x$ is less than 3". It says that 3 is less than $x$ , so $x > 3$ . When the unknown is on the right, swap the sides and turn the sign round with them.
Giving one number as the solution of an inequality. $x = 4$ is a solution of $x + 5 > 8$ , but the solution is $x > 3$ : every number greater than 3.
Counting the value that makes the two sides equal as a solution of $<$ or $>$ . When $x = 3$ , $2x + 1 > 7$ becomes $7 > 7$ , which is false. Only $\le$ and $\ge$ allow the two sides to be equal.
Mixing up "not more than" and "not less than". "Not more than 10" is $x \le 10$ . "Not less than 10" is $x \ge 10$ .
Counting 0 as a positive integer. 0 is a whole number, but it is neither positive nor negative.

Practice

Work each question on paper before you open its solution.

Question 1

Is each statement true or false?
(a) $-6 \le -6$
(b) $2 \ge 9$
(c) $-1 > -10$
(d) $-3 < -8$
(e) $0 \ge -0.5$
(f) $-\frac{2}{3} < -\frac{3}{4}$

ShowHide solution

(a) The two numbers are equal, and $\le$ allows "equal to".

(b) 2 is less than 9.

(d) $-3$ is to the right of $-8$ , so it is greater than $-8$ .

(e) 0 is greater than every negative number.

(f) $-\frac{2}{3} = -0.666\ldots$ and $-\frac{3}{4} = -0.75$ . Both are negative, and $-0.75$ is further from zero, so $-\frac{2}{3}$ is the greater.

(a) True (b) False (c) True (d) False (e) True (f) False

Question 2

Write each inequality in words. Then write it with the unknown on the left.
(a) $9 > y$
(b) $-1 \le m$
(c) $0 \ge t$
(d) $\frac{1}{2} < w$

ShowHide solution

(a) "9 is greater than $y$ ", so $y$ is less than 9.

(b) " $-1$ is less than or equal to $m$ ", so $m$ is greater than or equal to $-1$ .

(d) "A half is less than $w$ ", so $w$ is greater than a half.

(a) $y < 9$ (b) $m \ge -1$ (c) $t \le 0$ (d) $w > \frac{1}{2}$

Question 3

Explain the difference between
(a) $x < 10$ and $x \le 10$ ,
(b) the equation $x = 7$ and the inequality $x > 7$ .

ShowHide solution

(a) Both are satisfied by every number less than 10, such as 9.9, 9, 5 and $-100$ . The number 10 itself satisfies $x \le 10$ , because $10 \le 10$ is true, but not $x < 10$ , because $10 < 10$ is false.

(b) The equation $x = 7$ has one solution, 7. The inequality $x > 7$ has infinitely many, such as 7.1, 8 and 100, and 7 itself is not one of them.

(a) Only $x \le 10$ allows $x$ to be 10. (b) $x = 7$ has exactly one solution; $x > 7$ has infinitely many, and 7 is not one of them.

Question 4

Say whether each is an equation, an inequality or neither. For each equation and each inequality, say whether it has one solution or many.
(a) $4x = 20$
(b) $4x \le 20$
(c) $4x + 20$
(d) $7 > k$
(e) $x - 3 = -8$

ShowHide solution

(a) The sign $=$ : an equation. Only $x = 5$ makes it true.

(b) The sign $\le$ : an inequality. 5, 0 and $-3$ are three of its many solutions.

(d) The sign $>$ : an inequality. It says $k < 7$ , so 6, 0 and $-20$ are three of its many solutions.

(e) The sign $=$ : an equation. Only $x = -5$ makes it true, because $-5 - 3 = -8$ .

(a) An equation, with one solution (b) An inequality, with many solutions (c) Neither (d) An inequality, with many solutions (e) An equation, with one solution

Question 5

Is the given value a solution of the inequality?
(a) $x = 6$ ; $3x - 4 > 14$
(b) $x = -2$ ; $5 - x \le 7$
(c) $y = 0.5$ ; $4y + 1 < 3$
(d) $k = -2$ ; $2k + 9 \ge 4$

ShowHide solution

(a) $3(6) - 4 = 18 - 4 = 14\text{,}$ and $14 > 14$ is false.

(b) $5 - (-2) = 5 + 2 = 7\text{,}$ and $7 \le 7$ is true.

(d) $2(-2) + 9 = -4 + 9 = 5\text{,}$ and $5 \ge 4$ is true.

(a) No (b) Yes (c) No (d) Yes

Question 6

Of the numbers $-3$ , $-1$ , 0, 2, 2.5 and 5, which satisfy
(a) $x \ge 2$ ,
(b) $2x - 1 < 3$ ,
(c) $1 - x > -2$ ?

ShowHide solution

(a) $\ge$ allows 2 itself, and 2.5 and 5 are greater than 2.

(b) $x = -3$ : $2(-3) - 1 = -7$ , and $-7 < 3$ is true. $x = -1$ : $-3 < 3$ , true. $x = 0$ : $-1 < 3$ , true. $x = 2$ : $3 < 3$ , false. $x = 2.5$ : $4 < 3$ , false. $x = 5$ : $9 < 3$ , false.

(c) $x = -3$ : $1 - (-3) = 4$ , and $4 > -2$ is true. $x = -1$ : $2 > -2$ , true. $x = 0$ : $1 > -2$ , true. $x = 2$ : $-1 > -2$ , true. $x = 2.5$ : $-1.5 > -2$ , true. $x = 5$ : $-4 > -2$ , false.

(a) 2, 2.5 and 5 (b) $-3$ , $-1$ and 0 (c) $-3$ , $-1$ , 0, 2 and 2.5

Question 7

(a) $x$ is a positive integer and $x \le 5$ . List the possible values of $x$ .
(b) $x$ is a whole number and $x < 3$ . List the possible values of $x$ .
(c) $n$ is a positive integer and $4n \le 18$ . List the possible values of $n$ .
(d) How many whole numbers satisfy $x < 10$ ?

ShowHide solution

(a) The positive integers start at 1, and $\le$ allows 5 itself.

(b) The whole numbers start at 0, and 3 itself is not allowed.

(d) The whole numbers less than 10 are 0, 1, 2, …, 9.

(a) 1, 2, 3, 4, 5 (b) 0, 1, 2 (c) 1, 2, 3, 4 (d) 10 whole numbers

Question 8

Write an inequality for each sentence. Say what your letter stands for, with its unit.
(a) The mass of a cabin bag must not exceed 7 kg.
(b) More than 30 000 people watched a football match at the National Stadium.
(c) The temperature inside a freezer is below $-18$ °C.

ShowHide solution

(a) Let $m$ kg be the mass of the bag. "Must not exceed 7" allows 7 and anything less.

(b) Let $n$ be the number of people. "More than 30 000" does not allow 30 000 itself.

(a) $m \le 7$ (b) $n > 30\,000$ (c) $t < -18$

Question 9

When 7 is added to three times a number $n$ , the result is not more than 25.
(a) Write an inequality in $n$ .
(b) Is $n = 6$ a solution? Is $n = 7$ ?

ShowHide solution

(a) Three times the number is $3n$ , and adding 7 gives $3n + 7$ . "Not more than 25" allows 25 and anything less.

(b) When $n = 6$ : $3(6) + 7 = 25$ , and $25 \le 25$ is true. When $n = 7$ : $3(7) + 7 = 28$ , and $28 \le 25$ is false.

(a) $3n + 7 \le 25$ (b) $n = 6$ is a solution; $n = 7$ is not.

Question 10

Nurul has $15 on her card for bus fares. Each bus trip costs her $1.20.
(a) Write an inequality for $t$ , the number of trips she can pay for.
(b) Can she pay for 12 trips? Can she pay for 13?

ShowHide solution

(a) $t$ trips cost $1.2t$ dollars, and the cost must not exceed 15 dollars.

(b) When $t = 12$ : $1.2(12) = 14.4$ , and $14.4 \le 15$ is true. When $t = 13$ : $1.2(13) = 15.6$ , and $15.6 \le 15$ is false.

(a) $1.2t \le 15$ (b) Yes: 12 trips cost $14.40. No: 13 trips would cost $15.60, which is more than $15.

Question 11

To pass a test, a student needs at least 50 marks. Siti scored 38 marks in Section A, and she scores $m$ marks in Section B.
(a) Write an inequality that says that Siti passes.
(b) Which of these Section B marks are enough for her to pass: 10, 12, 15?
(c) What is the least whole number of marks she needs in Section B?

ShowHide solution

(a) Her total mark is $38 + m$ , and it must be 50 or more.

(b) When $m = 10$ : $38 + 10 = 48$ , and $48 \ge 50$ is false. When $m = 12$ : $50 \ge 50$ is true. When $m = 15$ : $53 \ge 50$ is true.

(a) $38 + m \ge 50$ (b) 12 and 15 (c) Siti needs at least 12 marks in Section B.

Key Points

An inequality compares two expressions with $<$ , $>$ , $\le$ or $\ge$ . $<$ and $>$ do not allow the two sides to be equal; $\le$ and $\ge$ do.
$3 < x$ says the same as $x > 3$ : swap the sides and turn the sign round with them.
A linear equation such as $x + 5 = 8$ has exactly one solution. An inequality such as $x + 5 > 8$ has infinitely many, written together as $x > 3$ .
To check whether a value is a solution, substitute it and decide whether the statement about the two numbers is true.
When the unknown counts things, list the whole-number or positive-integer solutions. 0 is a whole number, but not a positive integer.
To write an inequality from words, say what the letter stands for, with its unit, then choose the sign: "at least" and "not less than" give $\ge$ ; "at most", "not more than" and "must not exceed" give $\le$ .