Inequality Basics: Practice

The set is unbounded on the left and the right endpoint $3$ carries a bracket, so $3$ is included.

$(-\infty, 3] \quad\Longleftrightarrow\quad x \le 3$

The bracket means an inclusive symbol.

Divide both sides by $-2$. The divisor is negative, so reverse the direction from $<$ to $>$.

$\frac{-2x}{-2} > \frac{6}{-2} \;\Rightarrow\; x > -3$

Forgetting the flip would give the wrong half of the line.

Each end keeps its own circle: the strict left bound $-2$ is open, and the inclusive right bound $3$ is filled.

$-2 < x \le 3 \;\Rightarrow\; \text{open at } -2, \text{ filled at } 3$

The solution is the band of numbers between them.

A solution must be at least $-2$, meaning $-2$ or larger.

$-1 \ge -2 \;\text{ is true}$

The others, $-5$, $-3$, and $-10$, are all smaller than $-2$, so they fail.

Swap the sides and reverse the symbol to read the inequality from the other end.

$a \le b \quad\Longleftrightarrow\quad b \ge a$

Both say '$a$ is at most $b$.'

Multiply both sides by $4$. Multiplying by a positive number keeps the direction.

$4 \cdot \frac{x}{4} \ge 4 \cdot 2 \;\Rightarrow\; x \ge 8$

No flip occurs, because $4$ is positive.

This is transitivity: if $a$ is left of $b$ and $b$ is left of $c$, then $a$ is left of $c$.

$a < b \text{ and } b < c \;\Rightarrow\; a < c$

The comparisons chain together in one direction.

Add $7$ to both sides. Adding does not change the direction.

$x - 7 + 7 \ge -2 + 7 \;\Rightarrow\; x \ge 5$

The symbol stays as $\ge$.

The symbol $<$ is strict, so $20$ is excluded.

$x < 20 \;\Rightarrow\; \text{less than } 20$

'At most $20$' would be $x \le 20$, and 'at least' or 'no less than' would be $x \ge 20$.

Here $-x$ is $-1 \cdot x$, so divide both sides by $-1$. Dividing by a negative reverses the direction from $>$ to $<$.

$\frac{-x}{-1} < \frac{5}{-1} \;\Rightarrow\; x < -5$

The flip is required because the divisor is negative.

Both fences are parentheses, so both endpoints are excluded.

$(2, 7) \quad\Longleftrightarrow\quad 2 < x < 7$

The solution is every number strictly between $2$ and $7$.

The divisor $-4$ is negative, so reverse the direction from $\le$ to $\ge$, and $8 \div (-4) = -2$.

$\frac{-4x}{-4} \ge \frac{8}{-4} \;\Rightarrow\; x \ge -2$

Keeping $\le$ would graph the wrong half of the line.