Solve multi-step linear inequalities in one variable, represent the solution set on a number line and in interval notation, and interpret solutions in context, flipping the inequality when multiplying or dividing by a negative (A.EI.2).

What this topic is asking

A.EI.2 asks you to solve linear inequalities in one variable, show the solution on a number line (and in interval notation), and interpret it in context. On the Virginia Algebra I SOL these are Equations and Inequalities items: solve and type an inequality, pick the correct number-line graph, or place the circle and shading yourself with a hot-spot or coordinate tool.

Solve like an equation, with one new rule

To solve a linear inequality, isolate the variable using inverse operations, just as with an equation. The one difference is the flip rule:

When you multiply or divide both sides by a negative number, reverse the inequality sign.

For example, $-2x < 8$ becomes $x > -4$: dividing by $-2$ flips $<$ to $>$. Adding, subtracting, and multiplying or dividing by a positive number leave the sign unchanged.

Why the sign flips for a negative

The flip rule is not arbitrary. Multiplying by a negative number reflects every value across zero, which reverses their order on the number line. Concretely, $3 < 5$ is true, but multiplying both sides by $-1$ gives $-3$ and $-5$, and $-3 > -5$: the larger original became the smaller. So to keep the statement true after multiplying or dividing by a negative, the inequality must reverse. Testing a value after solving is the safest way to catch a missed flip.

Graphing on a number line

The graph of a one-variable inequality is a ray (or segment) on the number line:

• Open circle at the endpoint for strict inequalities ($<$, $>$): the endpoint is not a solution.
• Closed (filled) circle for inclusive inequalities ($\le$, $\ge$): the endpoint is a solution.
• Shade in the direction of all values that satisfy the inequality (right for "greater," left for "less," once the variable is alone on the left).

In interval notation, $x < -3$ is $(-\infty, -3)$ and $x \ge 2$ is $[2, \infty)$: a square bracket includes the endpoint, a parenthesis excludes it, and infinity always gets a parenthesis.

Inequalities in context

Word problems often produce an inequality and then need an interpretation. "You have \50$and each ticket costs$\$8$; how many can you buy?" gives $8t \le 50$, so $t \le 6.25$. Because you cannot buy a fraction of a ticket, the answer is at most 6 tickets: the context restricts the algebraic solution to whole numbers. Reading the situation (a count, a maximum, a minimum) tells you whether to round down, round up, or keep the full solution set.

How the SOL examines this topic

• Fill-in-the-blank. Solve and type the solution as an inequality such as $x \le -5$.
• Multiple choice. Match a number-line graph to an inequality, or pick the solution.
• Hot spot / coordinate tool. Place the open or closed circle and shade the correct direction on a number line.

A clarifying idea: an inequality has a range of solutions, not a single value, so its answer is a region of the number line. The endpoint type (open or closed) and the shading direction together capture that range.

Try this

Q1. Solve $3x - 7 < 11$. [1 point]

• Cue. $3x < 18 \Rightarrow x < 6$ (positive divide, no flip).

Q2. Solve $-5x \le 20$. [2 points]

• Cue. Divide by $-5$ and flip: $x \ge -4$.