2.2 Linear inequalities & compound inequalities

Linear Inequalities

An inequality compares two expressions that are not necessarily equal, using four symbols: < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). A linear inequality is solved almost exactly like a linear equation—use the same inverse operations to isolate the variable—with one critical difference you must never forget.

The One Rule That Changes Everything

When you multiply or divide both sides by a negative number, you must flip the inequality sign. Multiplying by a negative reverses the order of numbers on the number line: 3 < 5 is true, but multiplying both sides by -1 gives -3 and -5, and -3 > -5. Adding or subtracting never flips the sign—only multiplying or dividing by a negative does.

A helpful way to remember this: the flip only happens with the "strong" operations of multiplication and division, and only when the number you use is negative. If you ever forget, test your answer with a sample number—if it fails, the sign was pointed the wrong way.

Example 1: Solve x + 4 < 9. Subtract 4 from both sides: x < 5. No sign flip, because we only subtracted.

Example 2: Solve -3x ≥ 12. Divide both sides by -3 and flip ≥ to ≤: x ≤ -4. Check with x = -5: -3(-5) = 15 ≥ 12, which is true.

Example 3 (two-step): Solve 5 - 2x > 11. Subtract 5: -2x > 6. Divide by -2 and flip > to <: x < -3.

Graphing on a Number Line

Solutions to inequalities are ranges, shown on a number line:

• Use an open circle (○) for < or >—the endpoint is not included.
• Use a closed (filled) circle (●) for ≤ or ≥—the endpoint is included.
• Shade the arrow toward every number that satisfies the inequality.

For x ≤ -4, place a filled circle at -4 and shade to the left. For x < 5, place an open circle at 5 and shade to the left.

Interval Notation

Interval notation is a compact way to write a solution set:

• Parentheses ( ) exclude an endpoint; brackets [ ] include it.
• Infinity ∞ and negative infinity -∞ always take a parenthesis, because you can never reach them.

Compound Inequalities: "And"

A compound inequality joins two inequalities. An "and" statement requires both parts to be true at the same time—the overlap, or intersection. These are often written as a single three-part inequality.

Example 4: Solve -3 < 2x + 1 ≤ 7. Whatever you do, do it to all three parts. Subtract 1 from each part: -4 < 2x ≤ 6. Divide each part by 2 (positive, so no flip): -2 < x ≤ 3. The solution is every number between -2 and 3, excluding -2 and including 3. In interval notation that is (-2, 3]. On a number line: an open circle at -2, a filled circle at 3, shaded between them.

Example 5 (with a flip): Solve 1 ≤ 4 - x < 6. Subtract 4 from all three parts: -3 ≤ -x < 2. Multiply all parts by -1 and flip both signs: 3 ≥ x > -2. Rewriting in increasing order gives -2 < x ≤ 3, or (-2, 3].

Compound Inequalities: "Or"

An "or" statement requires at least one part to be true—the union. The two pieces usually point in opposite directions and cannot be combined into a single three-part inequality.

Example 6: Solve 2x - 1 < -5 or 3x + 2 ≥ 14.

• Left piece: 2x - 1 < -5, so add 1: 2x < -4, then divide by 2: x < -2.
• Right piece: 3x + 2 ≥ 14, so subtract 2: 3x ≥ 12, then divide by 3: x ≥ 4.

The solution is x < -2 or x ≥ 4. In interval notation this is (-∞, -2) ∪ [4, ∞), where ∪ means union. On a number line, shade left of an open circle at -2 and right of a filled circle at 4.

Putting It Together

Example 7: Solve -2(x - 3) ≥ 10. Distribute first: -2x + 6 ≥ 10. Subtract 6: -2x ≥ 4. Divide by -2 and flip: x ≤ -2. Interval notation: (-∞, -2].

Checking a Solution With a Test Point

Because inequalities have many answers, you verify them by testing one number from your solution set. For x ≤ -2, pick a value that should work, such as x = -3, and substitute it into the original inequality -2(x - 3) ≥ 10: -2(-3 - 3) = -2(-6) = 12, and 12 ≥ 10 is true. Then test a value that should fail, such as x = 0: -2(0 - 3) = 6, and 6 ≥ 10 is false. Because the "should-work" value passes and the "should-fail" value does not, the solution and the direction of the sign are both correct. This test-point habit is the fastest way to catch a forgotten flip.

Quick Reference

• Solve inequalities using the same inverse operations you use for equations.
• Flip the inequality sign only when multiplying or dividing by a negative number.
• Open circle = strict (<, >); filled circle = inclusive (≤, ≥).
• "And" = overlap (intersection); "or" = union of the two pieces.
• Infinity always gets a parenthesis in interval notation, never a bracket.