Integer Solutions of Inequalities

Two ends at once

$-3<x\le 5$−3 < x ≤ 5 is two inequalities written as one. The left part, −3 < x, says that x is greater than −3. The right part, $x\le 5$x ≤ 5, says that x is less than or equal to 5. Read the whole thing as "x is greater than −3 and at most 5".

A solution has to satisfy both parts. On the number line, one part shades everything to the right of −3 and the other shades everything up to 5. The solutions are where the two shadings overlap: the stretch between −3 and 5.

The stretch from −3 to 5. The circle at −3 is hollow and the circle at 5 is filled.

Which ends belong

Each end follows its own sign. At −3 the sign is <, and −3 < −3 is false, so −3 is not a solution and its circle is hollow. At 5 the sign is $\le$≤, and $5\le 5$5 ≤ 5 is true, so 5 is a solution and its circle is filled.

Listing the integers

Between −3 and 5 there are infinitely many numbers: −2.9, 0.5, 4.99 and every other number in the stretch. The integers are the whole numbers 0, 1, 2, 3 and so on, together with their negatives, and only a few of them lie in the stretch.

To list them, start at the first integer past the hollow end, which is −2, and count up to the filled end, which is 5: −2, −1, 0, 1, 2, 3, 4, 5.

The integers from −2 up to 5. The gold 5 is in because its end is filled; −3 is not marked, because its end is hollow.

Counting them

There are eight integers in the list. You can count them without listing: from −3 up to 5 is 5 − (−3) = 8 steps of 1. Each step lands on one integer, and the last step lands on 5. The start, −3, is not counted, so there are exactly 8.

The ends change the count, so check them first. $-3\le x\le 5$−3 ≤ x ≤ 5 lets −3 in as well, and has 9 integer solutions. −3 < x < 5 leaves both ends out, and has 7.

When an end is not an integer

Find the greatest integer x with $3x\le 14$3x ≤ 14. Dividing both sides by 3 gives $x\le \frac{14}{3}$x ≤ 14/3, which is $4\frac{2}{3}$4 2/3. The integers up to $4\frac{2}{3}$4 2/3 stop at 4, not 5, because 3 × 5 = 15 is more than 14. The greatest integer solution is 4, even though $4\frac{2}{3}$4 2/3 is nearer to 5 than to 4.

When neither end is an integer, hollow and filled make no difference to the list, because neither end could be on it. The integers with −2.5 < x < 3.2 are −2, −1, 0, 1, 2 and 3.

The usual mistakes

Counting the hollow end. For $-3<x\le 5$−3 < x ≤ 5, the list does not start at −3, because −3 is not greater than −3. The count is 8, not 9.

Dropping the filled end. $5\le 5$5 ≤ 5 is true, so 5 is on the list. Leaving it out gives 7.

Rounding to the nearest integer. $x\le 4\frac{2}{3}$x ≤ 4 2/3 allows 4 but not 5, so the answer is found by rounding down.