Inequalities: Practice

Add the constant $5$ to both sides, then divide by the positive coefficient $3$. The symbol stays.

$3x - 5 + 5 \ge 7 + 5 \;\Rightarrow\; 3x \ge 12 \;\Rightarrow\; x \ge 4$

Check the boundary: $x = 4$ gives $3(4) - 5 = 7 \ge 7$, which is true.

Subtract the constant $4$ to reach $\frac{x}{2} > 2$, then multiply by the positive number $2$. The symbol stays.

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

Check inside the range: $x = 6$ gives $\frac{6}{2} + 4 = 7 > 6$, which is true.

Subtract the constant $2$ to reach $5x \le -10$, then divide by the positive coefficient $5$. No flip occurs.

$5x \le -10 \;\Rightarrow\; x \le -2$

Check the boundary: $x = -2$ gives $5(-2) + 2 = -8 \le -8$, which is true.

Add the constant $3$ to both sides to reach $-4x \ge 12$, then divide by $-4$ and reverse the symbol.

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

Check the boundary: $x = -3$ gives $-4(-3) - 3 = 12 - 3 = 9 \ge 9$, which is true.

'At most $19$ dollars' means the cost $2m + 5$ is at most $19$, so $2m + 5 \le 19$. Subtract $5$, then divide by $2$.

$2m + 5 \le 19 \;\Rightarrow\; 2m \le 14 \;\Rightarrow\; m \le 7$

So you can ride at most $7$ miles.

Subtract the constant $4$ to reach $-3x < -9$, then divide by $-3$ and reverse the symbol.

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

Check: $x = 4$ gives $-3(4) + 4 = -8 < -5$, which is true.

Add the constant $1$ to reach $\frac{x}{4} \ge 3$, then multiply by the positive number $4$. The symbol stays.

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

Check the boundary: $x = 12$ gives $\frac{12}{4} - 1 = 3 - 1 = 2 \ge 2$, which is true.

Solve first: subtract $1$ to get $2x \ge 6$, then divide by $2$ to get $x \ge 3$. The symbol $\ge$ includes the boundary, so use a closed circle, and 'greater' shades right.

$x \ge 3 \;\Rightarrow\; \text{closed circle on } 3, \text{ shade right}$

Graph D matches; the others use the wrong circle or shade left.

'More than $11$' means $3n - 4 > 11$. Add $4$, then divide by the positive coefficient $3$.

$3n - 4 > 11 \;\Rightarrow\; 3n > 15 \;\Rightarrow\; n > 5$

Check: $n = 6$ gives $3(6) - 4 = 14 > 11$, which is true.

Subtract the constant $2$ to reach $-5x > 10$, then divide by $-5$ and reverse the symbol.

$-5x > 10 \;\Rightarrow\; x < \frac{10}{-5} = -2$

Check: $x = -3$ gives $-5(-3) + 2 = 17 > 12$, which is true.

Subtract the constant $7$ to reach $3x < -6$, then divide by the positive coefficient $3$. No flip occurs, even though the answer is negative.

$3x < -6 \;\Rightarrow\; x < -2$

Check: $x = -3$ gives $3(-3) + 7 = -2 < 1$, which is true.

Add the constant $1$ to reach $-\frac{x}{2} > 3$, then multiply both sides by $-2$ and reverse the symbol.

$-\frac{x}{2} > 3 \;\Rightarrow\; x < 3 \cdot (-2) = -6$

Check: $x = -8$ gives $-\frac{-8}{2} - 1 = 4 - 1 = 3 > 2$, which is true.