Inequalities

One-step inequalities (addition and subtraction)

To solve the inequality we need to isolate the variable (a letter used as a placeholder for an unknown value, mostly x or y) to one side and everything else to the other side. All inequalities have two sides: a left side (LS) and a right side (RS). The relationship between sides can be < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).

We can to do the same thing (adding, subtracting, multiplying every term, dividing every term) to both sides, what can help to bring like terms (numbers) together to one side and isolate the variable on the other side.

Example:

$$8−x < 3x$$

$$8 < 3x + x$$

$$8 < 4x$$

$$8÷4 < x$$

$$2 < x$$

One-step inequalities (multiplication and division)

Inequalities

To solve the inequality we need to isolate the variable (a letter used as a placeholder for an unknown value, mostly x or y) to one side and everything else to the other side. All inequalities have two sides: a left side (LS) and a right side (RS). The relationship between sides can be < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).

We can to do the same thing (adding, subtracting, multiplying every term, dividing every term) to both sides, what can help to bring like terms (numbers) together to one side and isolate the variable on the other side.

Example:

$$8−x < 3x$$

$$8 < 3x + x$$

$$8 < 4x$$

$$8÷4 < x$$

$$2 < x$$

Choose the inequality represented by a graph

Draw inequality

We display inequalities as lines on the number axis. If the endpoint is a solid dot, it is ≤ (less than or equal to), ≥ (greater than or equal to), and if it is hollow, the sign is <,>.

So for −2

Choose the interval represented by a graph

Interval notation

An Interval notation is a method of writing down a set/range of numbers. We need brackets and a pair of numbers representing two endpoints of a number range. For this we use brackets [] and parentheses () for this. Brackets [ ] mean that the endpoint of the range is included (this is drawn on a number line with a solid dot). Parentheses () mean that the endpoint is excluded and does not contain the listed element (this is drawn on a number line with hollow dot).

So for (−2; 3>, the range starts after −2 (−2 is excluded), but ends with 3 (including 3).

We can write this as:

$$ −2 < x ≤ 3$$

Or draw:

Infinity symbols ∞ and −∞ are always accompanied by round brackets (). For example, [2, ∞) is the interval of real numbers greater or equal to 2.