Inequalities: Additional Practice (Set 2)

Subtract the constant $3$ from both sides, then divide by the positive coefficient $2$. Neither step flips the symbol.

$2x + 3 - 3 < 11 - 3 \;\Rightarrow\; 2x < 8 \;\Rightarrow\; x < 4$

Check: $x = 0$ gives $2(0) + 3 = 3 < 11$, which is true.

Subtract the constant $5$ from both sides, then divide by the positive coefficient $4$. Neither step flips the symbol.

$4x + 5 - 5 \le 21 - 5 \;\Rightarrow\; 4x \le 16 \;\Rightarrow\; x \le 4$

Check the boundary: $x = 4$ gives $4(4) + 5 = 21 \le 21$, which is true.

Subtract the constant $1$ from both sides; subtraction does not move the symbol. Then divide by $-2$ and reverse it.

$-2x + 1 - 1 > 7 - 1 \;\Rightarrow\; -2x > 6 \;\Rightarrow\; x < -3$

Check: $x = -4$ gives $-2(-4) + 1 = 9 > 7$, which is true.

Test the value by substituting it into the original inequality, with the order of operations.

$-3(-1) + 2 = 3 + 2 = 5, \qquad 5 \ge 4 \;\checkmark$

Since $5 \ge 4$ is true, $x = -1$ is a solution. (Solving gives $x \le -\tfrac{2}{3}$, and $-1 \le -\tfrac{2}{3}$, which agrees.)

Subtract the constant $4$ from both sides to reach $-3x \le 9$. Then divide by $-3$ and reverse the symbol.

$-3x \le 9 \;\Rightarrow\; x \ge \frac{9}{-3} = -3$

Check: $x = 0$ gives $4 - 3(0) = 4 \le 13$, which is true.

Distribute the $-3$ to get $-3x - 3 > 6$. Add $3$ to both sides, then divide by $-3$ and reverse the symbol.

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

Check: $x = -4$ gives $-3(-4 + 1) = -3(-3) = 9 > 6$, which is true.

Subtract the constant $7$ to reach $-2x > -6$, then divide by $-2$ and reverse the symbol.

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

Check: $x = 0$ gives $7 - 2(0) = 7 > 1$, which is true.

Add $x$ to both sides to gather the variable on the right, reaching $3 \le 4x - 5$. Then add $5$ and divide by the positive number $4$.

$3 \le 4x - 5 \;\Rightarrow\; 8 \le 4x \;\Rightarrow\; 2 \le x$

Reading $2 \le x$ the other way gives $x \ge 2$. Adding $x$ keeps the symbol, since it is an addition.

Distribute the $2$ across both terms inside the parentheses, then solve the two-step inequality.

$2(x - 1) = 2x - 2 \le 8 \;\Rightarrow\; 2x \le 10 \;\Rightarrow\; x \le 5$

Check the boundary: $x = 5$ gives $2(5 - 1) = 2(4) = 8 \le 8$, which is true.

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

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

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

Subtract the constant $1$ to reach $-2x \ge 8$, then divide by $-2$ and reverse the symbol.

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

Check the boundary: $x = -4$ gives $1 - 2(-4) = 1 + 8 = 9 \ge 9$, which is true.

Distribute the $3$ to get $3x + 6 > x + 8$. Subtract $x$ from both sides, then subtract $6$.

$3x + 6 > x + 8 \;\Rightarrow\; 2x + 6 > 8 \;\Rightarrow\; 2x > 2 \;\Rightarrow\; x > 1$

Check: $x = 2$ gives $3(4) = 12$ and $2 + 8 = 10$, and $12 > 10$ is true.