Solve linear inequalities in one variable, including those requiring the distributive property and those with variables on both sides, graph the solution, and interpret it in context (TEKS A.5B).

What this topic is asking

TEKS A.5B is the inequality companion to solving equations, in the same heavily weighted Reporting Category 3. You solve a one-variable inequality almost exactly like an equation, with one extra rule, then graph the solution on a number line and interpret it in context. The redesigned test often makes the graph itself the answer through a hot-spot item.

Solve like an equation, with one rule

The steps mirror equation-solving: distribute, combine like terms, collect the variable, and divide. The one difference is the sign-flip rule.

For $4x - 1 < 11$: add 1 to get $4x < 12$, divide by positive 4 (no flip) to get $x < 3$. For $-5x \le 20$: divide by $-5$ and flip to get $x \ge -4$.

Graphing the solution on a number line

The solution to a one-variable inequality is a ray (or interval) on the number line.

• Circle style. Open circle for a strict inequality ($<$ or $>$); closed circle for $\le$ or $\ge$.
• Direction. Shade toward the solutions: right for $>$ or $\ge$, left for $<$ or $\le$, once the variable is alone on the left.

So $x > 28$ is an open circle at 28 with a ray to the right; $x \le -4$ is a closed circle at $-4$ with a ray to the left.

Variables on both sides

Collect the variable on one side first, then apply the rule only if the final division is by a negative.

Interpreting in context

In a real-world inequality, finish with a sentence. A solution $m > 5$ months means "from month 6 onward"; a solution $s \le 20$ items means "at most 20 items". Where only whole numbers make sense, state the smallest or largest acceptable whole number, the interpretive credit the standard rewards.

How STAAR examines this topic

• Multiple choice. Solve an inequality; the "forgot to flip" answer is always offered.
• Hot spot. Build the number-line graph (circle style plus ray direction), both of which must be correct.
• Inline choice and number entry. State a boundary value or choose the correct symbol.

A clarifying idea is why the sign flips: multiplying by a negative reverses the order of numbers (for example $2 < 5$ but $-2 > -5$), so to keep the statement true the symbol must turn around.

Compound inequalities and "at least / at most" language

Some inequalities have two bounds, written as a compound statement. "Between 10 and 20 inclusive" is $10 \le x \le 20$, graphed as a segment with closed circles at both ends. You solve each part the same way, applying any operation to all three regions at once: from $-1 \le 2x - 3 \le 5$, add 3 throughout to get $2 \le 2x \le 8$, then divide by 2 to get $1 \le x \le 4$.

Translating the everyday phrasing is its own assessed skill. "At least" means $\ge$, "at most" means $\le$, "more than" means $>$, and "no more than" means $\le$. A budget that allows spending "no more than \$50" becomes a$\le 50$ constraint, and the solution describes every affordable amount. Reading these phrases as the correct symbol is where many context inequalities are won or lost, because the rest of the algebra is routine once the symbol is right.

Try this

Q1. Solve $-4x + 3 < 19$. [1 point]

• Cue. $-4x < 16 \Rightarrow x > -4$ (flip on dividing by $-4$).

Q2. Graph $x \ge 2$ on a number line. [1 point]

• Cue. Closed circle at 2, ray to the right.