Inequalities: When Equals Is Not Enough

Equations find points. Inequalities find regions.

That's the whole difference. When you solve x + 3 = 7, you get x = 4 — a single location on the number line. When you solve x + 3 > 7, you get x > 4 — an entire territory. Every number bigger than 4. An infinite set.

Here's why this matters: most real constraints aren't equalities. "Be at least this tall." "Spend no more than this much." "Arrive before this time." The world runs on boundaries and ranges, not exact values. Inequalities describe what's possible, not what's required.

So instead of asking "what is x?" inequalities ask "what can x be?" The answer isn't a point. It's a region of points.

The Inequality Symbols

< means "less than": x < 5 means x is smaller than 5 > means "greater than": x > 5 means x is larger than 5 ≤ means "less than or equal to": x ≤ 5 includes 5 itself ≥ means "greater than or equal to": x ≥ 5 includes 5 itself

The symbol points to the smaller side. Think of it as an alligator mouth eating the bigger number.

Solving Inequalities (Almost Like Equations)

Most steps work just like equations. Whatever you do to one side, do to the other.

Example: x + 3 > 7

Subtract 3 from both sides: x > 4

Example: 2x ≤ 10

Divide both sides by 2: x ≤ 5

So far, same rules as equations.

The Big Exception: Multiplying or Dividing by Negatives

Here's where inequalities differ from equations:

When you multiply or divide by a negative number, flip the inequality sign.

Why? Multiplying by -1 reverses order:

3 < 5, but -3 > -5

The smaller number becomes the larger one on the negative side.

Example: -2x > 6

Divide by -2 (negative), so flip the sign: x < -3

Check: Is -4 in the solution? -2(-4) = 8. Is 8 > 6? Yes ✓ Is -2 in the solution? -2(-2) = 4. Is 4 > 6? No, so -2 shouldn't be in solution ✓

Why the Flip Happens

Consider 4 > 2.

Multiply both sides by -1: -4 and -2

Is -4 > -2? No. -4 < -2.

Negation reverses order. The number that was bigger becomes smaller (more negative). So the inequality direction must flip to stay true.

Graphing Solutions on a Number Line

Inequalities have solutions you can visualize:

x > 4: Open circle at 4, shade everything to the right. (Open circle = 4 not included)

x ≤ 4: Closed circle at 4, shade everything to the left. (Closed circle = 4 included)

The shaded region represents all numbers in the solution set.

Compound Inequalities

Sometimes you have two conditions at once:

"And" (intersection): Both conditions must be true.

2 < x ≤ 5 means x > 2 AND x ≤ 5

x is between 2 and 5 (not including 2, including 5).

"Or" (union): At least one condition must be true.

x < 2 or x > 5

x is less than 2, or greater than 5 (the middle is excluded).

Solving Compound Inequalities

For "and" inequalities, solve both parts:

3 < 2x + 1 ≤ 9

Subtract 1 from all three parts: 2 < 2x ≤ 8

Divide by 2: 1 < x ≤ 4

Solution: x is between 1 and 4, not including 1, including 4.

Absolute Value Inequalities

Absolute value measures distance from zero. |x| is "how far x is from 0."

|x| < 3: Distance from zero is less than 3. Solution: -3 < x < 3 (between -3 and 3)

|x| > 3: Distance from zero is greater than 3. Solution: x < -3 or x > 3 (outside the interval)

Pattern:

• |expression| < k becomes -k < expression < k
• |expression| > k becomes expression < -k or expression > k

Solving Absolute Value Inequalities

Example: |2x - 1| < 5

Convert: -5 < 2x - 1 < 5

Add 1: -4 < 2x < 6

Divide by 2: -2 < x < 3

Example: |x + 2| ≥ 4

Convert: x + 2 ≤ -4 or x + 2 ≥ 4

Solve: x ≤ -6 or x ≥ 2

Quadratic Inequalities

What about x² - 4 > 0?

First, solve the equation x² - 4 = 0. x² = 4 x = ±2

These are the boundary points. They divide the number line into regions.

Test each region:

• x < -2: Try x = -3. (-3)² - 4 = 9 - 4 = 5 > 0. ✓
• -2 < x < 2: Try x = 0. 0² - 4 = -4 < 0. ✗
• x > 2: Try x = 3. 3² - 4 = 5 > 0. ✓

Solution: x < -2 or x > 2

Systems of Inequalities

Multiple inequalities at once define a region in the plane.

y > x y < 3

Graph both:

• y > x: everything above the line y = x
• y < 3: everything below the line y = 3

The solution is the intersection: the region that satisfies both.

This is how linear programming works — optimizing within a region defined by multiple constraints.

Interval Notation

Mathematicians often write solutions as intervals:

(2, 5] means 2 < x ≤ 5

• Parenthesis ( or ) means endpoint NOT included
• Bracket [ or ] means endpoint included

[1, 4) means 1 ≤ x < 4

(-∞, 3) means x < 3 (all numbers less than 3)

[0, ∞) means x ≥ 0 (all non-negative numbers)

Real-World Inequalities

Equations describe exact relationships. Inequalities describe constraints and bounds.

"You must be at least 48 inches tall to ride." → height ≥ 48

"Speed limit is 65 mph." → speed ≤ 65

"Budget is under $500." → cost < 500

Real problems rarely have exact requirements — they have minimums, maximums, and ranges.

The Core Insight

Equations find points. Inequalities find regions.

When you solve an inequality, you're not finding the answer. You're finding all the answers — every value that makes the statement true.

The techniques are almost the same as equations, with one crucial difference: multiply or divide by a negative, and the inequality flips.

Inequalities describe what's possible, not what's exact. They're about boundaries and ranges, minimums and maximums — constraints that shape what can be.

Part 7 of the Algebra Fundamentals series.

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