Inequality Basics: Practice

The symbol $>$ means 'greater than', and it points from the larger number to the smaller one.

$7 > 4 \;\Rightarrow\; \text{seven is greater than four}$

The pointed end sits on the $4$, the smaller number.

'At most $10$' means $10$ is allowed and nothing larger, which is 'less than or equal to'.

$n \le 10$

The inclusive symbol keeps $10$ in the solution set.

The symbol $>$ is strict, so $x$ must be larger than $6$, not equal to it.

$6 > 6 \;\text{ is false}$

Since $6$ is not greater than itself, $6$ is not a solution.

The symbol is strict, so the boundary $4$ is excluded and gets an open circle. 'Greater than' shades toward the larger numbers, to the right.

$x > 4 \;\Rightarrow\; \text{open circle on } 4, \text{ shade right}$

A filled circle would wrongly include $4$.

To turn an inequality end for end, swap the sides and reverse the symbol.

$x \ge 5 \quad\Longleftrightarrow\quad 5 \le x$

Both say '$x$ is at least $5$.'

The circle on $1$ is open, so $1$ is excluded, which means a strict symbol. The shading runs right toward the larger numbers, so the relation is 'greater than'.

$\text{open circle on } 1, \text{ shade right} \;\Rightarrow\; x > 1$

A filled circle would give $x \ge 1$, and shading left would give a 'less than' relation.

The boundary $2$ is excluded, so it takes a parenthesis, and the set runs right to infinity, which always takes a parenthesis.

$x > 2 \quad\Longleftrightarrow\quad (2, \infty)$

A bracket at $2$ would include it, and a bracket next to $\infty$ is never allowed.

The relation runs left from $0$, so the left end is $-\infty$ with a parenthesis, and the included boundary $0$ takes a bracket.

$x \le 0 \quad\Longleftrightarrow\quad (-\infty, 0]$

The parenthesis form $(-\infty, 0)$ would exclude $0$.

A filled (closed) circle marks the boundary as part of the solution, matching an inclusive symbol $\le$ or $\ge$.

$\text{filled circle} \;\Rightarrow\; \text{boundary included}$

An open circle would mark it as excluded.

Infinity is a direction, not a number you can reach or include, so the interval stays open at that end.

$(2, \infty) \quad\text{and}\quad (-\infty, 5]$

Every infinite end takes a parenthesis, never a bracket.

'At least $12$' allows exactly $12$ and anything heavier, so it includes the boundary.

$w \ge 12$

The inclusive symbol keeps $12$ in the solution.

A solution set collects every value that makes the statement true, and here that is all numbers below $8$.

$x < 8 \;\Rightarrow\; \text{all numbers less than } 8$

There are infinitely many, which is why we draw a ray rather than a single dot.