Inequalities: Additional Practice (Set 1)

Add the constant $7$ to both sides to reach $-2x > 10$, then divide by $-2$ and reverse the symbol.

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

Check: $x = -6$ gives $-2(-6) - 7 = 12 - 7 = 5 > 3$, which is true.

Distribute the $5$ across both terms, then solve the two-step inequality.

$5(x - 2) = 5x - 10 \ge 5 \;\Rightarrow\; 5x \ge 15 \;\Rightarrow\; x \ge 3$

Check the boundary: $x = 3$ gives $5(3 - 2) = 5(1) = 5 \ge 5$, which is true.

Subtract the constant $8$ to reach $-5x < -15$, then divide by $-5$ and reverse the symbol.

$-5x < -15 \;\Rightarrow\; x > \frac{-15}{-5} = 3$

Check: $x = 4$ gives $8 - 5(4) = 8 - 20 = -12 < -7$, which is true.

Subtract the constant $2$ to reach $\frac{x}{3} \le 3$, then multiply by the positive number $3$. The symbol stays.

$\frac{x}{3} \le 3 \;\Rightarrow\; x \le 9$

Check the boundary: $x = 9$ gives $\frac{9}{3} + 2 = 3 + 2 = 5 \le 5$, which is true.

An open circle means the boundary is excluded, so the symbol is strict. Shading to the right means the solutions are larger than the boundary.

$\text{open circle on } -2, \text{ shade right} \;\Rightarrow\; x > -2$

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

Subtract the constant $6$ to reach $-x \ge 4$, then multiply by $-1$ and reverse the symbol.

$-x \ge 4 \;\Rightarrow\; x \le -4$

Check the boundary: $x = -4$ gives $-(-4) + 6 = 4 + 6 = 10 \ge 10$, which is true.

Gather the variable on one side: subtract $2x$ from both sides to reach $-9 \le 3x$. Then divide by the positive number $3$.

$-9 \le 3x \;\Rightarrow\; -3 \le x$

Reading $-3 \le x$ the other way gives $x \ge -3$. Subtracting $2x$ keeps the symbol, since it is a subtraction.

Subtract the constant $1$ to reach $\frac{x}{-4} \ge 2$, then multiply both sides by $-4$ and reverse the symbol.

$\frac{x}{-4} \ge 2 \;\Rightarrow\; x \le 2 \cdot (-4) = -8$

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

Solve first: divide $-x < 2$ by $-1$ and reverse the symbol, giving $x > -2$. The symbol is strict, so use an open circle, and 'greater' shades right.

$x > -2 \;\Rightarrow\; \text{open circle on } -2, \text{ shade right}$

Graph A matches; the others use a closed circle or shade left.

Subtract $4x$ from both sides to gather the variable, then solve the two-step inequality. Subtraction does not flip the symbol.

$7x - 2 > 4x + 7 \;\Rightarrow\; 3x - 2 > 7 \;\Rightarrow\; 3x > 9 \;\Rightarrow\; x > 3$

Check: $x = 4$ gives $7(4) - 2 = 26$ and $4(4) + 7 = 23$, and $26 > 23$ is true.

The variable is divided by $-3$ (the expression is $\frac{x}{-3}$), so multiply both sides by $-3$ and reverse the symbol.

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

Check the boundary: $x = -6$ gives $-\frac{-6}{3} = 2 \ge 2$, which is true.

The total weight is $2b + 4$, and 'at most $12$' means $2b + 4 \le 12$. Subtract $4$, then divide by $2$.

$2b + 4 \le 12 \;\Rightarrow\; 2b \le 8 \;\Rightarrow\; b \le 4$

So the box holds at most $4$ books.