Inequality Review Problems Study Guide

Inequality Review Problems Worksheet

Algebraic Isolation Rules:
- Treat the inequality symbol like an equals sign for standard algebraic steps (addition, subtraction, multiplication, division).
- Critical Rule: When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality symbol must be reversed.
Number Line Representation Rules:
- Open Circle ( $<$ or $>$ ): Used when the endpoint value is not included in the solution set.
- Solid / Closed Circle ( $\le$ or $\ge$ ): Used when the endpoint value is included in the solution set.
- Direction of Arrow:
- Point to the right (towards $\infty$ ) for greater than ( $>$ or $\ge$ ).
- Point to the left (towards $-\infty$ ) for less than ( $<$ or $\le$ ).
Interval Notation Rules:
- Use parentheses ( or ) for non-inclusive endpoints ( $<$ or $>$ ) and for infinite bounds ( $\infty$ or $-\infty$ ).
- Use square brackets [ or ] for inclusive endpoints ( $\le$ or $\ge$ ).

Original Inequality: $10(m - 3) > -4m + 6(m - 3)$
Step-by-Step Solution:
- Apply the distributive property to expand both sides: $10m - 30 > -4m + 6m - 18$
- Combine like terms on the right side ( $-4m + 6m = 2m$ ): $10m - 30 > 2m - 18$
- Subtract $2m$ from both sides: $8m - 30 > -18$
- Add $30$ to both sides: $8m > 12$
- Divide both sides by $8$ : $m > \frac{12}{8}$
- Simplify the fraction by dividing by $4$ : $m > \frac{3}{2}$ or $m > 1.5$
Solution Set: $m > 1.5$
Interval Notation: $(1.5, \infty)$
Graphing on Number Line:
- Open circle at $1.5$ (or $\frac{3}{2}$ ).
- Arrow extends to the right towards positive infinity.

Original Inequality: $-3(-1 + 4k) - 2 \ge -8k - 31$
Step-by-Step Solution:
- Distribute $-3$ across the terms inside parentheses: $3 - 12k - 2 \ge -8k - 31$
- Combine constant terms on the left side ( $3 - 2 = 1$ ): $1 - 12k \ge -8k - 31$
- Add $12k$ to both sides: $1 \ge 4k - 31$
- Add $31$ to both sides: $32 \ge 4k$
- Divide both sides by $4$ : $8 \ge k$ or $k \le 8$
Solution Set: $k \le 8$
Interval Notation: $(-\infty, 8]$
Graphing on Number Line:
- Solid (closed) circle at $8$ .
- Arrow extends to the left towards negative infinity.

Original Inequality: $-21 + 8x \ge -5(7x - 6) - 8x$
Step-by-Step Solution:
- Distribute $-5$ across the terms inside parentheses on the right side: $-21 + 8x \ge -35x + 30 - 8x$
- Combine like variable terms on the right side ( $-35x - 8x = -43x$ ): $-21 + 8x \ge -43x + 30$
- Add $43x$ to both sides: $-21 + 51x \ge 30$
- Add $21$ to both sides: $51x \ge 51$
- Divide both sides by $51$ : $x \ge 1$
Solution Set: $x \ge 1$
Interval Notation: $[1, \infty)$
Graphing on Number Line:
- Solid (closed) circle at $1$ .
- Arrow extends to the right towards positive infinity.

Original Inequality: $-1 + 7b \le -(-8b + 8)$
Step-by-Step Solution:
- Distribute the negative sign across the expression on the right side: $-1 + 7b \le 8b - 8$
- Subtract $7b$ from both sides: $-1 \le b - 8$
- Add $8$ to both sides: $7 \le b$ or $b \ge 7$
Solution Set: $b \ge 7$
Interval Notation: $[7, \infty)$
Graphing on Number Line:
- Solid (closed) circle at $7$ .
- Arrow extends to the right towards positive infinity.

Original Inequality: $\frac{x + 8}{12} < -1$
Step-by-Step Solution:
- Multiply both sides of the inequality by $12$ to clear the fraction: $x + 8 < -12$
- Subtract $8$ from both sides: $x < -20$
Solution Set: $x < -20$
Interval Notation: $(-\infty, -20)$
Graphing on Number Line:
- Open circle at $-20$ .
- Arrow extends to the left towards negative infinity.

Original Inequality: $-1 + \frac{x}{6} \ge -4$
Step-by-Step Solution:
- Add $1$ to both sides of the inequality: $\frac{x}{6} \ge -3$
- Multiply both sides by $6$ to solve for $x$ : $x \ge -18$
Solution Set: $x \ge -18$
Interval Notation: $[-18, \infty)$
Graphing on Number Line:
- Solid (closed) circle at $-18$ .
- Arrow extends to the right towards positive infinity.