Solve linear inequalities in one variable, graph the solution on a number line, and interpret the solution set in context (A.PAR, Patterning and Algebraic Reasoning).

What this topic is asking

This Patterning and Algebraic Reasoning (A.PAR) standard extends equation solving to inequalities in one variable. The method is almost identical to solving an equation, with one crucial difference: the inequality sign reverses whenever you multiply or divide both sides by a negative number. On the Georgia Milestones EOC, inequalities appear as numeric or hot-spot items where you solve and then represent the solution on a number line, and as multiple-choice items that translate a phrase like "at most" or "at least" into the correct inequality. Reading those phrases correctly and remembering the sign flip are the two skills being tested.

Solving like an equation, with one exception

To solve an inequality, isolate the variable using the same steps as for an equation: distribute, clear fractions, collect like terms, and divide. The one rule that differs:

To see why the flip is needed, note that $4 > 2$ is true, but multiplying both sides by $-1$ gives $-4$ and $-2$, and $-4 < -2$, so the direction must reverse to keep the statement true.

Graphing the solution on a number line

The solution set is a ray (or segment) on the number line.

• Open circle at the endpoint for strict inequalities $<$ or $>$ (the endpoint is not a solution).
• Closed circle at the endpoint for $\le$ or $\ge$ (the endpoint is a solution).
• Shade toward the values that satisfy the inequality: right for "greater than," left for "less than."

For $x \ge -3$: closed circle at $-3$, shade right. For $x < 4$: open circle at 4, shade left.

Translating words into inequalities

EOC word problems use phrases that map to specific symbols.

• "at most," "no more than," "maximum" map to $\le$.
• "at least," "no less than," "minimum" map to $\ge$.
• "more than," "greater than" map to $>$.
• "fewer than," "less than" map to $<$.

A budget of "at most $30" with a$4 base and $2 per mile gives$4 + 2m \le 30$, so$m \le 13$ miles.

How the Milestones examines this topic

• Numeric or hot-spot. Solve an inequality and select the correct number-line graph (open versus closed circle, correct direction).
• Multiple choice. Match a context phrase to the correct inequality, with sign-direction distractors.
• Constructed response. Solve a multi-step inequality and interpret the solution set in context, stating units.

Why the endpoint type matters in context

The open-versus-closed-circle distinction is not just notation; it changes the answer to a real question. If a ride costs "at most $30," the inequality is$4 + 2m \le 30$, so$m \le 13$, and 13 miles exactly is allowed, a **closed** circle, because$4 + 2(13) = 30$hits the budget exactly. But if the problem said "less than$30," then 13 miles would cost exactly 30 and would not qualify, so the endpoint is excluded with an open circle and the largest whole-mile answer becomes 12. Reading whether the boundary value itself is allowed is exactly the interpretive step the EOC rewards, and it is the difference between $\le$ and $<$.

Try this

Q1. Solve $3x - 7 > 5$ and describe the graph. [1 point]

• Cue. $3x > 12$, so $x > 4$; open circle at 4, shade right.

Q2. Solve $-\frac{x}{2} \ge 3$. [1 point]

• Cue. Multiply by $-2$ and flip: $x \le -6$.