Solve linear inequalities in one variable, flip the inequality when multiplying or dividing by a negative, and represent the solution as an interval and on a number line (Ohio A-REI.3).

What this topic is asking

Ohio standard A-REI.3 also covers linear inequalities in one variable. You solve them almost exactly like equations, with one extra rule, and then you describe the answer as a solution set: an interval, an inequality, or a graph on a number line. This skill is in the Expressions and Equations reporting category and shows up on Part 1.

Solving like an equation, with one twist

Everything you do to solve an equation works for an inequality, except the flip rule.

Showing the solution on a number line

The two choices are the dot type and the arrow direction.

• Dot type: filled (closed) for $\leq$ or $\geq$ because the endpoint is a solution; hollow (open) for $<$ or $>$ because it is not.
• Arrow direction: toward larger numbers for $>$ or $\geq$ (right), toward smaller numbers for $<$ or $\leq$ (left), once the variable is alone on the left.

So $x < 2$ is an open dot at $2$ with the arrow left, and $x \geq -1$ is a closed dot at $-1$ with the arrow right.

How Ohio examines this topic

• Equation/numeric response. Type the solution inequality, for example $x < -4$.
• Multiple choice. Match an inequality to its number-line graph, testing the dot type and direction.
• Drag and drop / graphing. Place the dot and arrow on a number line.

Watch the flip rule on every divide-by-negative; it is the single most-tested trap here.

Why dividing by a negative flips the sign

The flip rule can feel arbitrary, so it helps to see why it must be true. Start with a true inequality of plain numbers, say $2 < 5$. Multiply both sides by $-1$: the numbers become $-2$ and $-5$, and now $-2 > -5$, because $-2$ sits to the right of $-5$ on the number line. Multiplying by a negative reflects every number across zero, which reverses their order, so the inequality must reverse too to stay true. The same reflection happens whenever you multiply or divide by any negative, which is exactly when you flip. Tying the rule to this picture makes it something you can rederive under pressure instead of a fact you might misremember.

Reading the solution set in context

When an inequality models a situation, the solution set often needs a real-world reading. "You can spend at most $50$" becomes $\leq 50$, and a solution like $x \leq 12$ might mean "you can buy up to $12$ items." The endpoint's inclusion matters: "at most" and "no more than" include the endpoint ($\leq$), while "fewer than" and "under" exclude it ($<$). Translating the words to the right symbol, and back again, is part of the modeling that the test threads through this strand, and it connects directly to creating inequalities from context.

Try this

Q1. Solve $-5x + 2 < 17$ and describe the graph. [2 points]

• Cue. $-5x < 15$, divide by $-5$ and flip: $x > -3$; open dot at $-3$, arrow right.

Q2. Solve $2x - 1 \geq 7$. [1 point]

• Cue. $2x \geq 8$, so $x \geq 4$.