Integer Solutions of Inequalities

How to find the largest or smallest integer that satisfies an inequality, the largest or smallest prime, even, odd or square number among its solutions, and how to list and count whole-number solutions.

Before you start

The integers are …, $-3$ , $-2$ , $-1$ , 0, 1, 2, 3, … The positive integers are 1, 2, 3, … and the whole numbers are 0, 1, 2, 3, …

The prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, … and the perfect squares are 1, 4, 9, 16, 25, 36, …

From lessons 1.3 and 1.4: $2x + 1 < 9$ gives $2x < 8$ , so $x < 4$ . Dividing both sides by a negative number reverses the sign.

The Largest and the Smallest Integer

The solution $x < 6$ contains infinitely many numbers, but only some of them are integers: 5, 4, 3, 2, … The largest of them is 5, not 6, because 6 itself is not a solution. The solution $x \le 6$ contains 6, so its largest integer is 6.

Finding the largest or the smallest integer

Step 1: Solve the inequality.

Step 2: If the boundary value is an integer and the sign is $\le$ (for the largest) or $\ge$ (for the smallest), the boundary value itself is the answer. When the sign is $<$ , the largest integer is one less than the boundary value; when it is $>$ , the smallest integer is one more.

Step 3: If the boundary value is not an integer, the largest integer is the first integer to its left on the number line, and the smallest integer is the first integer to its right.

Example 1: The largest integer

Find the largest integer $x$ that satisfies
(a) $3x - 7 < 11$ ,
(b) $3x - 7 \le 11$ .

Solution

(a) Add 7 to both sides: $3x < 18$ . Divide both sides by 3: $x < 6$ .

The circle at 6 is open, so 6 is not a solution. The integers that the line covers are 5, 4, 3, …, and the largest is 5. Check: $3(5) - 7 = 8$ , and $8 < 11$ is true; $3(6) - 7 = 11$ , and $11 < 11$ is false.

(b) In the same way, $x \le 6$ . Now 6 is a solution: $11 \le 11$ is true.

(a) 5 (b) 6

Example 2: Negative and decimal boundary values

(a) Find the smallest integer $x$ that satisfies $-2x + 1 < 8$ .
(b) Find the largest integer $x$ that satisfies $5x + 2 \le 20$ .
(c) Find the smallest integer $x$ that satisfies $4 - x \le 9$ .

Solution

(a) Subtract 1 from both sides: $-2x < 7$ . Divide both sides by $-2$ , which reverses the sign: $x > -3.5$ .

The first integer to the right of $-3.5$ is $-3$ . $-4$ is to the left of $-3.5$ , so it is not a solution: $-2(-4) + 1 = 9$ , and $9 < 8$ is false.

(b) Subtract 2 from both sides: $5x \le 18$ . Divide by 5: $x \le 3.6$ . The first integer to the left of 3.6 is 3. Rounding 3.6 to 4 would be wrong: $5(4) + 2 = 22$ , which is more than 20.

(c) Subtract 4 from both sides: $-x \le 5$ . Multiply both sides by $-1$ , which reverses the sign: $x \ge -5$ . The boundary value $-5$ is an integer and is included.

(a) $-3$ (b) 3 (c) $-5$

Primes, Even and Odd Numbers, Squares and Multiples

A question may ask for the largest or smallest number of a certain kind. Solve the inequality first, then go along the integers from the boundary value until you reach one of that kind.

Example 3: Numbers of a certain kind

(a) $x$ satisfies $3x - 6 \le 70$ . Find the largest integer, the largest prime number, the largest even number, the largest perfect square and the largest multiple of 7 that $x$ can be.
(b) $y$ satisfies $5y + 3 > 40$ . Find the smallest integer, the smallest prime number, the smallest odd number and the smallest multiple of 6 that $y$ can be.

Solution

(a) Add 6 to both sides: $3x \le 76$ . Divide by 3: $x \le 25\frac{1}{3}$ .

Going down from $25\frac{1}{3}$ : the integers are 25, 24, 23, … 25 is the largest integer, and it is also a perfect square ( $5^{2}$ ). 24 is the largest even number. 25 is not prime ( $5 \times 5$ ) and 24 is not, but 23 is. The largest multiple of 7 is 21.

(b) Subtract 3 from both sides: $5y > 37$ . Divide by 5: $y > 7.4$ .

Going up from 7.4: the integers are 8, 9, 10, 11, 12, … 8 is the smallest integer. 9 is the smallest odd number. 8, 9 and 10 are not prime, but 11 is. The smallest multiple of 6 is 12.

(a) Integer: 25; prime: 23; even number: 24; perfect square: 25; multiple of 7: 21 (b) Integer: 8; prime: 11; odd number: 9; multiple of 6: 12

Listing and Counting Whole-Number Solutions

Example 4: Listing and counting

(a) List the positive integers $x$ that satisfy $2x + 3 \le 15$ , and say how many there are.
(b) How many whole numbers $n$ satisfy $17 - 3n > 2$ ?
(c) How many positive odd integers satisfy $4x - 1 < 30$ ?

Solution

(a) Subtract 3 from both sides: $2x \le 12$ . Divide by 2: $x \le 6$ . The positive integers start at 1, and 6 is included.

(b) Subtract 17 from both sides: $-3n > -15$ . Divide by $-3$ , which reverses the sign: $n < 5$ . The whole numbers start at 0, and 5 is not included: 0, 1, 2, 3, 4.

(c) Add 1 to both sides: $4x < 31$ . Divide by 4: $x < 7.75$ . The positive odd integers less than 7.75 are 1, 3, 5 and 7.

(a) 1, 2, 3, 4, 5, 6: six positive integers (b) 5 whole numbers (c) 4 positive odd integers

Integer Solutions from Words

The inequality may be given in words. Write it with the letter first, choosing the sign from the words, then solve it and read the integers off the solution as before.

Example 5: An inequality in words

When 5 is added to three times a whole number $n$ , the result is less than 30.
(a) Write an inequality in $n$ and solve it.
(b) Find the largest possible value of $n$ .
(c) How many whole numbers $n$ are possible?

Solution

(a) Three times the number is $3n$ , and adding 5 gives $3n + 5$ . "Less than 30" does not allow 30 itself: $3n + 5 < 30$ . Subtract 5 from both sides: $3n < 25$ . Divide both sides by 3: $n < 8\frac{1}{3}$ .

(b) The first integer to the left of $8\frac{1}{3}$ is 8. Check: $3(8) + 5 = 29$ , and $29 < 30$ is true; $3(9) + 5 = 32$ , and $32 < 30$ is false.

(a) $3n + 5 < 30$ , so $n < 8\frac{1}{3}$ (b) 8 (c) 9 whole numbers, 0 to 8

Common Mistakes

Taking the boundary value when it is not included. The largest integer with $x < 6$ is 5, and the smallest integer with $x > 4$ is 5.
Rounding a boundary value to the nearest integer. The largest integer with $x \le 3.6$ is 3, not 4: in Example 2 (b), $5(4) + 2 = 22$ , which is more than 20.
Going the wrong way from a negative boundary value. The smallest integer with $x > -3.5$ is $-3$ . $-4$ is less than $-3.5$ , so it is not a solution.
Starting the list at the wrong number. The positive integers start at 1 and the whole numbers at 0: $n < 5$ has five whole-number solutions, 0 to 4, but only four positive-integer solutions, 1 to 4.

Practice

Work each question on paper before you open its solution.

Question 1

Find the largest integer $x$ that satisfies
(a) $4x + 3 < 23$ ,
(b) $7 - 2x \ge -4$ .

ShowHide solution

(a) Subtract 3: $4x < 20$ . Divide by 4: $x < 5$ . 5 is not included, so the largest integer is 4.

(b) Subtract 7: $-2x \ge -11$ . Divide by $-2$ , which reverses the sign: $x \le 5.5$ . The first integer to the left of 5.5 is 5.

(a) 4 (b) 5

Question 2

Find the smallest integer $x$ that satisfies
(a) $3x - 2 > 10$ ,
(b) $-4x < 18$ ,
(c) $2(x + 5) \ge 5$ .

ShowHide solution

(a) Add 2: $3x > 12$ . Divide by 3: $x > 4$ . 4 is not included, so the smallest integer is 5.

(b) Divide by $-4$ , which reverses the sign: $x > -4.5$ . The first integer to the right of $-4.5$ is $-4$ .

(c) Expand: $2x + 10 \ge 5$ . Subtract 10: $2x \ge -5$ . Divide by 2: $x \ge -2.5$ . The first integer to the right of $-2.5$ is $-2$ .

(a) 5 (b) $-4$ (c) $-2$

Question 3

Find
(a) the largest integer $x$ that satisfies $5x + 12 < -13$ ,
(b) the smallest integer $x$ that satisfies $2 - 5x \le 30$ ,
(c) the largest integer $x$ that satisfies $\frac{x}{3} + 2 \le 1$ .

ShowHide solution

(a) Subtract 12 from both sides: $5x < -25$ . Divide both sides by 5: $x < -5$ . $-5$ is not included, so the largest integer is one less: $-6$ .

(b) Subtract 2 from both sides: $-5x \le 28$ . Divide both sides by $-5$ , which reverses the sign: $x \ge -5.6$ . The first integer to the right of $-5.6$ is $-5$ .

(c) Subtract 2 from both sides: $\frac{x}{3} \le -1$ . Multiply both sides by 3: $x \le -3$ . $-3$ is an integer and is included.

(a) $-6$ (b) $-5$ (c) $-3$

Question 4

Hui Min says that the smallest integer that satisfies $-3x < 10$ is $-4$ . Is she right? Explain.

ShowHide solution

Divide both sides by $-3$ , which reverses the sign: $x > -\frac{10}{3}$ , that is, $x > -3\frac{1}{3}$ .

The solutions are to the right of $-3\frac{1}{3}$ . $-4$ is to its left, so it is not a solution: $-3(-4) = 12$ , and $12 < 10$ is false. The first integer to the right of $-3\frac{1}{3}$ is $-3$ : $-3(-3) = 9$ , and $9 < 10$ is true.

No. The smallest integer is $-3$ . Hui Min went to the left of $-3\frac{1}{3}$ , but the solutions are to its right.

Question 5

$x$ satisfies $4x - 9 \le 50$ . Find
(a) the largest integer,
(b) the largest prime number,
(c) the largest multiple of 4,
(d) the largest perfect square
that $x$ can be.

ShowHide solution

Add 9: $4x \le 59$ . Divide by 4: $x \le 14.75$ . The integers are 14, 13, 12, …

(a) 14. (b) 14 is not prime ( $2 \times 7$ ), but 13 is. (c) 12. (d) The perfect squares are 1, 4, 9, 16, …, and 16 is too large, so 9.

(a) 14 (b) 13 (c) 12 (d) 9

Question 6

$y$ satisfies $3y + 1 > 20$ . Find
(a) the smallest integer,
(b) the smallest even number,
(c) the smallest perfect square,
(d) the smallest multiple of 5
that $y$ can be.

ShowHide solution

Subtract 1: $3y > 19$ . Divide by 3: $y > 6\frac{1}{3}$ . The integers are 7, 8, 9, 10, …

(a) 7. (b) 8. (c) 9, which is $3^{2}$ . (d) 10.

(a) 7 (b) 8 (c) 9 (d) 10

Question 7

(a) List the positive integers $x$ that satisfy $5x - 4 \le 18$ .
(b) How many whole numbers $n$ satisfy $2(n - 3) < 9$ ?

ShowHide solution

(a) Add 4: $5x \le 22$ . Divide by 5: $x \le 4.4$ . The positive integers up to 4.4 are 1, 2, 3 and 4.

(b) Expand: $2n - 6 < 9$ . Add 6: $2n < 15$ . Divide by 2: $n < 7.5$ . The whole numbers are 0, 1, 2, 3, 4, 5, 6 and 7.

(a) 1, 2, 3, 4 (b) 8 whole numbers

Question 8

(a) How many positive integers satisfy $7x - 5 \le 60$ ?
(b) How many of them are odd?

ShowHide solution

Add 5 to both sides: $7x \le 65$ . Divide both sides by 7: $x \le 9\frac{2}{7}$ .

(a) The positive integers start at 1, and the largest one in the solution is 9: they are 1, 2, 3, …, 9.

(b) The odd ones are 1, 3, 5, 7 and 9.

(a) 9 positive integers (b) 5 of them are odd

Question 9

In 6 years' time, Mei's age will be less than twice her age now. Her age now is a whole number of years. What is the youngest she can be now?

ShowHide solution

Let Mei be $a$ years old now. In 6 years she will be $a + 6$ years old, and twice her age now is $2a$ : $a + 6 < 2a$ .

Subtract $a$ from both sides: $6 < a$ , which is $a > 6$ .

6 is not included, so the smallest whole number is 7. Check: at 7, she will be 13, and $13 < 14$ is true. At 6, she would be 12, and $12 < 12$ is false.

The youngest Mei can be now is 7 years old.

Question 10

A positive integer $n$ is divided by 3, and 4 is added to the result. The answer is less than 9.
(a) Find the largest possible value of $n$ .
(b) How many values of $n$ are possible?

ShowHide solution

$\frac{n}{3} + 4 < 9$ . Subtract 4 from both sides: $\frac{n}{3} < 5$ . Multiply both sides by 3: $n < 15$ .

(a) 15 is not included, so the largest integer is 14. Check: $\frac{14}{3} + 4 = 8\frac{2}{3}$ , which is less than 9; $\frac{15}{3} + 4 = 9$ , which is not.

(b) The positive integers less than 15 are 1, 2, 3, …, 14.

(a) 14 (b) 14 values

Key Points

Solve the inequality first, then read the integers off its solution.
When the boundary value is an integer, it is the largest integer for $\le$ and the smallest for $\ge$ . For $<$ the largest integer is one less than it; for $>$ the smallest integer is one more.
When the boundary value is not an integer, the largest integer is the first integer to its left and the smallest is the first integer to its right. Never round to the nearest: the largest integer in $x \le 3.6$ is 3.
With a negative boundary value, go along the number line, not towards or away from zero: the smallest integer in $x > -3.5$ is $-3$ , the first integer to the right of $-3.5$ .
For a number of a certain kind (prime, even, odd, square, multiple), go along the integers from the boundary value until you reach one.
To list or count solutions, see where the list starts (1 for positive integers, 0 for whole numbers) and whether the boundary value is included.