Solve linear inequalities in one variable and represent the solution on a number line, applying the sign-flip rule for negatives (NC.M1.A-REI.3).

What this topic is asking

NC.M1.A-REI.3 includes solving linear inequalities in one variable. The procedure is the same as for equations, with one extra rule: multiplying or dividing both sides by a negative reverses the inequality sign. You then represent the solution on a number line and interpret it in context.

The flip rule

The only new idea versus equations is the sign flip.

Why does it flip? Because multiplying by a negative reverses order on the number line: $2 < 3$ is true, but $-2 < -3$ is false; the correct statement is $-2 > -3$.

A solving routine with a flip

Testing one value from your solution set in the original inequality is the fastest way to catch a missed flip.

Open versus closed circles

The endpoint marker depends on the symbol:

• Open circle for $<$ or $>$: the endpoint is not a solution.
• Closed (filled) circle for $\le$ or $\ge$: the endpoint is a solution.

Then shade the direction the variable can go: $x > a$ and $x \ge a$ shade right; $x < a$ and $x \le a$ shade left.

Reading an inequality written backward

Inequalities are sometimes written with the variable on the right, which trips students up. A statement like $7 \ge 2x + 1$ means exactly the same as $2x + 1 \le 7$: reading it as "$7$ is greater than or equal to $2x + 1$" is identical to "$2x + 1$ is at most $7$." You can either solve it in place ($6 \ge 2x$, so $3 \ge x$, that is $x \le 3$) or flip the whole statement around first so the variable is on the left. Both routes give $x \le 3$. The safe habit is to rewrite with the variable on the left before graphing, so the shading direction reads naturally from the symbol.

How the NC Math 1 EOC examines this topic

• Multiple choice. Solve and choose the correct graph (circle type and shading direction).
• Technology-enhanced. Place the circle and shade the ray on a number-line tool.
• Calculator-inactive. Inequality solving is core no-calculator fluency.

This topic feeds directly into creating inequalities from context and into graphing inequalities in two variables, where the boundary becomes a line and the solution becomes a half-plane.

Why an inequality has a region of solutions

An equation pins the variable to specific values; an inequality opens it to a range. Solving $x > 5$ does not find "the answer," it describes the entire set of numbers greater than $5$. This is why the solution is a shaded ray, not a point, and why checking a single test value verifies the whole set: every number on the correct side behaves the same way. Holding this picture, a solution as a region rather than a number, prevents the common error of reporting a single value for an inequality and prepares you for two-variable inequalities, where the region becomes a half-plane.

Try this

Q1. Solve $-5x \le 20$. [2 points]

• Cue. Divide by $-5$ and flip: $x \ge -4$ (closed circle at $-4$, shade right).

Q2. Solve $3x + 2 < 11$. [1 point]

• Cue. $3x < 9 \Rightarrow x < 3$ (no flip; open circle at $3$, shade left).