12.2 Linear Inequalities

12.2 Linear Inequalities

Quick Answer: Solve a linear inequality like an equation—isolate the variable with inverse operations—except reverse the inequality symbol whenever you multiply or divide both sides by a negative number. Graph with an open circle for strict inequalities ($<$, $>$) and a closed circle for inclusive ones ($\le$, $\ge$), then shade the true side.

Linear inequalities are a named PERT Math competency. You will solve one-variable inequalities, read solution sets from number-line graphs, and occasionally handle simple compound statements. The algebra is close to linear equations; the direction of the inequality is what trips people up.

Inequality Symbols at a Glance

Inclusive vs exclusive: $\le$ and $\ge$ include the boundary point (filled circle). $<$ and $>$ exclude it (open circle).

Solving One-Step and Multi-Step Linear Inequalities

Properties That Stay the Same as Equations

You may:

• Add the same number to both sides
• Subtract the same number from both sides
• Multiply or divide both sides by the same positive number

The inequality symbol keeps its direction in all of those cases.

The Critical Difference: Negatives

If you multiply or divide both sides by a negative number, reverse the inequality:

• $<$ becomes $>$
• $>$ becomes $<$
• $\le$ becomes $\ge$
• $\ge$ becomes $\le$

Why? Multiplying by −1 reflects the number line. If $2 < 5$ is true, then $-2 > -5$ after multiplying by −1. The order flips. The same reflection happens when you divide by a negative.

You do not reverse when you add or subtract a negative—only when you multiply or divide by a negative.

Worked Example 1: Basic Isolation

Problem: Solve $3x - 7 < 8$.

Step 1: Add 7 to both sides: $3x < 15$.

Step 2: Divide by 3 (positive—do not reverse): $x < 5$.

Graph: Open circle at 5, shade left.

Check: Try $x = 0$: $3(0) - 7 = -7 < 8$ true. Try $x = 6$: $3(6) - 7 = 11 < 8$ false. Boundary $x = 5$ is excluded because the symbol is strict.

Worked Example 2: Reverse When Dividing by a Negative

Problem: Solve $-2x + 5 \ge 13$.

Step 1: Subtract 5: $-2x \ge 8$.

Step 2: Divide by −2 and reverse $\ge$ to $\le$:

$x \le -4$

Graph: Closed circle at −4, shade left (all numbers less than or equal to −4).

Check: $x = -4$: $-2(-4) + 5 = 8 + 5 = 13 \ge 13$ true. $x = -5$: $-2(-5) + 5 = 10 + 5 = 15 \ge 13$ true. $x = 0$: $5 \ge 13$ false. Direction is correct.

Trap: Students often get $x \ge -4$ by forgetting to reverse. Always pause when a negative coefficient sits in front of the variable.

Worked Example 3: Variables on Both Sides

Problem: Solve $5x - 3 > 2x + 9$.

Step 1: Subtract $2x$: $3x - 3 > 9$.

Step 2: Add 3: $3x > 12$.

Step 3: Divide by 3: $x > 4$.

No reverse was needed because the divisor was positive. Collecting variable terms on one side and constants on the other mirrors equation solving.

Worked Example 4: Distribute First

Problem: Solve $2(3 - x) \le 10$.

Step 1: Distribute: $6 - 2x \le 10$.

Step 2: Subtract 6: $-2x \le 4$.

Step 3: Divide by −2 and reverse: $x \ge -2$.

Alternatively, after $6 - 2x \le 10$, some students move $-2x$ by adding $2x$ and subtracting 10: $6 - 10 \le 2x \Rightarrow -4 \le 2x \Rightarrow -2 \le x$, which is the same as $x \ge -2$. This path avoids dividing by a negative if you keep the variable positive. Use whichever path reduces your personal error rate.

Graphing on a Number Line

Circle Rules

Interval Notation (Useful on Answer Choices)

Parentheses mean the endpoint is excluded; brackets mean included. Infinity always gets a parenthesis: never $[3, \infty]$.

Compound Inequalities Basics

PERT may include simple compounds. Know the two logical joins:

“And” (Intersection)

Both conditions must hold. Example: $x > 1$ and $x < 4$ is written $1 < x < 4$. On the number line, shade only the overlap. Graph: open circles at 1 and 4, shade between them.

Solving a three-part chain: $ -3 \le 2x + 1 < 7 $.

Step 1: Subtract 1 from all three parts: $-4 \le 2x < 6$.

Step 2: Divide all three parts by 2: $-2 \le x < 3$.

Interval: $[-2, 3)$.

If you multiply or divide a chain by a negative, reverse both inequality symbols.

“Or” (Union)

At least one condition holds. Example: $x \le -1$ or $x > 2$. Shade left of −1 (closed) and right of 2 (open). There is no single closed interval between them; the solution is a union of two rays.

Trap: Treating “or” as if you need the middle segment, or treating “and” as if either side alone is enough.

Worked Example 5: Compound “And” Chain

Problem: Solve $-1 < 3 - 2x \le 7$.

Step 1: Subtract 3 from all parts: $-4 < -2x \le 4$.

Step 2: Divide by −2 and reverse both symbols: $2 > x \ge -2$, which rewrites neatly as $-2 \le x < 2$.

Check: $x = 0$: $-1 < 3 \le 7$ true. $x = 2$: $-1 < 3 - 4 \le 7 \Rightarrow -1 < -1 \le 7$ false (strict left inequality fails). $x = -2$: $-1 < 3 - 2(-2) \le 7 \Rightarrow -1 < 7 \le 7$ true.

Special Solution Sets

Occasionally simplification produces a statement with no variable:

• All real numbers: After simplifying, you get a always-true statement such as $0 < 5$. Solution: $(-\infty, \infty)$ or “all real $x$.”
• No solution: You get a never-true statement such as $0 > 2$. Solution: empty set $\emptyset$.

These appear less often than standard rays, but knowing them prevents panic if options include “no solution” or “all real numbers.”

Connecting Inequalities to Word Language

PERT may phrase constraints in words:

Translate the English carefully before algebra. “At least” and “no less than” include equality; “more than” does not.

Strategy Card for PERT Inequality Items

1. Write the inequality cleanly; distribute and combine like terms.
2. Isolate the variable; mark any step that multiplies/divides by a negative.
3. Reverse the symbol on those negative multiplications/divisions only.
4. Express the answer as inequality, graph description, or interval—matching the choices.
5. Test one number from your solution set and one outside it in the original inequality.

PERT-Style Traps

1. Forgetting to reverse when dividing by a negative coefficient.
2. Reversing when adding/subtracting a negative (incorrect).
3. Wrong circle: closed circle for $<$ or open for $\le$.
4. Shading the wrong side after a reverse.
5. Flipping only one symbol in a three-part compound when dividing by a negative.
6. Confusing “and” with “or” on compound items.

Quick Mental Checks

• After $x < a$, numbers smaller than $a$ should work.
• After $x \ge a$, the boundary $a$ must satisfy the original.
• If you reversed once, your boundary solution to the related equation is still a boundary—but the true side of the number line swapped.

Linear inequalities reward the same careful inverse operations as equations, plus one habit: name the step that multiplies or divides by a negative and reverse the symbol there. Pair that habit with open/closed circle discipline, and this competency becomes highly reliable points on PERT Math.