Solving Linear Inequalities: Practice

Collect the variable on the left by subtracting $2x$ from both sides, then finish with the constants.

$3x - 4 > 8 \;\Rightarrow\; 3x > 12 \;\Rightarrow\; x > 4$

Dividing by the positive number $3$ keeps the direction.

Collect the variable on the right, where the coefficient stays positive, by subtracting $3x$ from both sides.

$7 \le 2x - 1 \;\Rightarrow\; 8 \le 2x \;\Rightarrow\; 4 \le x$

Reading $4 \le x$ from the other end gives $x \ge 4$, with no flip needed.

Distribute, then collect the variable on the left.

$6x - 3 \ge 2x + 5 \;\Rightarrow\; 4x \ge 8 \;\Rightarrow\; x \ge 2$

Dividing by the positive number $4$ keeps the symbol as $\ge$.

Subtract $1$ from all three parts, then divide all three by the positive number $2$.

$-2 \le 2x < 6 \;\Rightarrow\; -1 \le x < 3$

The left end is inclusive (bracket) and the right end is strict (parenthesis), giving $[-1, 3)$.

Subtract $4x$ from both sides, then the constants.

$3x - 2 > 13 \;\Rightarrow\; 3x > 15 \;\Rightarrow\; x > 5$

Dividing by the positive number $3$ keeps the direction.

Add $x$ to both sides so the variable lands on the right with a positive coefficient, avoiding a flip.

$4 \ge 3x - 5 \;\Rightarrow\; 9 \ge 3x \;\Rightarrow\; 3 \ge x$

Reading $3 \ge x$ from the other end gives $x \le 3$.

Distribute on both sides, then collect the variable on the left.

$5x - 10 < 3x + 12 \;\Rightarrow\; 2x < 22 \;\Rightarrow\; x < 11$

Dividing by the positive number $2$ keeps the direction.

Divide all three parts by the positive number $3$.

$\frac{0}{3} \le \frac{3x}{3} < \frac{9}{3} \;\Rightarrow\; 0 \le x < 3$

The left end is inclusive (bracket) and the right end is strict (parenthesis), giving $[0, 3)$.

Subtract $8$ from both sides to get $-x > -5$, then divide by $-1$ and reverse the symbol.

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

The flip comes from dividing by a negative number.

Subtract $2x$ from both sides so the variable stays on the right with a positive coefficient.

$6 \le 4x - 2 \;\Rightarrow\; 8 \le 4x \;\Rightarrow\; 2 \le x$

Reading from the other end gives $x \ge 2$, with no flip needed.

Multiply both sides by $-2$. Because the multiplier is negative, reverse $<$ to $>$.

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

The flip comes from multiplying by a negative number.

Add $4$ to all three parts, then divide all three by the positive number $2$.

$-2 \le 2x \le 6 \;\Rightarrow\; -1 \le x \le 3$

Both ends are inclusive, so both take brackets, giving $[-1, 3]$.