Solve linear inequalities in one variable and represent the solution on a number line and in interval form, reversing the inequality when multiplying or dividing by a negative (TN A1.A.REI.B.3).

What this topic is asking

Standard A1.A.REI.B.3 also covers inequalities. Solving an inequality is almost identical to solving an equation, with one extra rule: multiplying or dividing both sides by a negative number reverses the inequality sign. You then express the solution as an inequality, on a number line (open or closed circle plus a shaded ray), or in interval notation.

The flip rule

Adding or subtracting from both sides never changes the direction. Multiplying or dividing by a negative does. The reason is concrete: $2 < 5$ is true, but multiplying both sides by $-1$ gives $-2$ and $-5$, and $-2 > -5$, so the direction must reverse to stay true.

Graphing on a number line

The two graph decisions are the circle type and the direction.

• Open circle for $<$ or $>$: the endpoint is not a solution.
• Closed (filled) circle for $\le$ or $\ge$: the endpoint is a solution.
• Shade toward the solutions: $x > a$ shades right; $x < a$ shades left.

A reliable check is to pick a test value in the shaded region and confirm it satisfies the original inequality.

Compound inequalities

A compound inequality combines two conditions. An "and" inequality like $-2 < x \le 5$ is the overlap (a segment between $-2$ open and $5$ closed). An "or" inequality like $x < 1$ or $x \ge 4$ is two separate rays. To solve a three-part inequality such as $-1 \le 2x + 3 < 7$, apply each operation to all three parts at once.

How TNReady examines this topic

• Multiple choice. Solve and choose the correct inequality, with unflipped-sign distractors.
• Graphing. Place the circle (open or closed) and shade the ray on a number line.
• Multiple select. Choose all values that satisfy the inequality.

A clarifying idea is that an inequality has infinitely many solutions (a whole region), unlike a linear equation's single value. That is why the answer is a graph or a range, and why "select all that apply" items suit inequalities so well.

Why the circle type matters for credit

On a graphing item, the open-versus-closed circle is scored, and it encodes a real distinction. For $x > 3$, the value $x = 3$ gives $3 > 3$, which is false, so $3$ is excluded and the circle is open. For $x \ge 3$, the value $x = 3$ gives $3 \ge 3$, which is true, so $3$ is included and the circle is filled. Getting the boundary right is not a formality: in a context problem, $\ge$ versus $>$ can be the difference between "at least 3 items" (3 allowed) and "more than 3 items" (3 not allowed). Read the inequality symbol and translate it faithfully to the circle.

Interval notation and writing the answer

Some TNReady items accept or display the solution in interval notation, a compact way to write a range. A bracket $[\,$ or $]\,$ includes the endpoint (matches a closed circle); a parenthesis $(\,$ or $)\,$ excludes it (matches an open circle), and $\infty$ always takes a parenthesis because infinity is never reached. So $x > 3$ is $(3, \infty)$, $x \le -3$ is $(-\infty, -3]$, and the compound $-2 < x \le 5$ is $(-2, 5]$. Reading between the three forms, inequality, number line, and interval, is a small but testable skill, and the bracket-versus-parenthesis choice carries the same open-or-closed information as the circle.

Try this

Q1. Solve $-5x < 20$. [1 point]

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

Q2. Solve and graph $3x + 1 \le 7$. [2 points]

• Cue. $3x \le 6 \Rightarrow x \le 2$; closed circle at $2$, shade left.