Inequalities: Practice

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

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

Check: $x = -5$ gives $-2(-5) = 10 > 8$, true; keeping $>$ would graph the wrong side.

The variable is multiplied by $-3$, so divide both sides by $-3$. The divisor is negative, so reverse the symbol from $\le$ to $\ge$.

$\frac{-3x}{-3} \ge \frac{12}{-3} \;\Rightarrow\; x \ge -4$

Dividing by a bare $3$ or keeping $\le$ gives the wrong half of the line.

Divide both sides by $-5$. The divisor is negative, so reverse the symbol from $\ge$ to $\le$.

$\frac{-5x}{-5} \le \frac{15}{-5} \;\Rightarrow\; x \le -3$

Check: $x = -4$ gives $-5(-4) = 20 \ge 15$, which is true.

Divide both sides by $-4$. The divisor is negative, so reverse the symbol from $<$ to $>$. A negative divided by a negative is positive.

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

Check: $x = 3$ gives $-4(3) = -12 < -8$, which is true.

Test a value by substituting it into the original inequality, before any rearranging.

$-3(0) = 0, \qquad 0 < 6 \;\checkmark$

Since $0 < 6$ is true, $x = 0$ is a solution. (Solving gives $x > -2$, and $0 > -2$, which agrees.)

Add $4$ to both sides. Adding does not change the symbol, and the variable can stay on the right.

$3 + 4 > x - 4 + 4 \;\Rightarrow\; 7 > x$

Reading $7 > x$ from the other end gives $x < 7$, the same statement.

Divide both sides by $-2$. The divisor is negative, so reverse the symbol from $\le$ to $\ge$. A negative divided by a negative is positive.

$\frac{-2x}{-2} \ge \frac{-10}{-2} \;\Rightarrow\; x \ge 5$

Check: $x = 6$ gives $-2(6) = -12 \le -10$, which is true.

Subtract $9$ from both sides. Subtraction does not change the symbol, even though the answer is negative.

$x + 9 - 9 < 3 - 9 \;\Rightarrow\; x < -6$

A negative result does not trigger a flip; only multiplying or dividing by a negative does. Check: $x = -7$ gives $-7 + 9 = 2 < 3$, true.

Subtract $12$ from both sides. Subtracting does not change the symbol.

$7 - 12 < x + 12 - 12 \;\Rightarrow\; -5 < x$

Reading $-5 < x$ from the other end gives $x > -5$, the same statement.

Divide both sides by $-3$. The divisor is negative, so reverse the symbol from $>$ to $<$. A negative divided by a negative is positive.

$\frac{-3x}{-3} < \frac{-9}{-3} \;\Rightarrow\; x < 3$

Check: $x = 0$ gives $-3(0) = 0 > -9$, which is true.

The variable is divided by $-2$, so multiply both sides by $-2$. Multiplying by a negative reverses the symbol from $\le$ to $\ge$.

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

Check: $x = 0$ gives $\frac{0}{-2} = 0 \le 4$, true, and $0 \ge -8$ holds.

Divide both sides by $-2$. The divisor is negative, so reverse the symbol.

$\frac{10}{-2} \le \frac{-2x}{-2} \;\Rightarrow\; -5 \le x$

Reading $-5 \le x$ from the other end gives $x \ge -5$, the same statement.