Solving Inequalities and Graphing Zeros

What You're Actually Dealing With

Inequalities and zeros are two sides of the same coin. Zeros tell you where a function hits the x-axis. Inequalities tell you where the function is above or below that axis. If you're struggling with either, you're probably mixing these concepts or skipping the fundamentals.

Let's fix that.

Understanding Zeros First

A zero of a function is simply where y = 0. On a graph, that's where the curve crosses the x-axis. Finding zeros means solving f(x) = 0.

For a linear equation like f(x) = 2x - 6, you set it equal to zero:

2x - 6 = 0
2x = 6
x = 3

The zero is x = 3. That's it.

For quadratic equations, you factor, use the quadratic formula, or complete the square. The method changes based on what you're working with.

Quick Zero-Finding Methods

• Linear equations: Isolate x. One solution max.
• Quadratic equations: Factor if possible. Use quadratic formula if factoring fails. You get zero, one, or two solutions.
• Higher-degree polynomials: Factor by grouping or synthetic division. More complex, but the goal stays the same.

Solving Inequalities: The Rules

Inequalities follow most algebra rules, but one twist breaks people: the direction flips when you multiply or divide by a negative number.

That's the whole gotcha. Remember it or get wrong answers every time.

Linear Inequalities

Solve 3x + 5 < 14:

3x + 5 < 14
3x < 9
x < 3

No negative division happened, so the sign stays the same. The solution is x < 3.

Now solve -2x + 7 > 3:

-2x + 7 > 3
-2x > -4
x < 2

See what happened? Dividing by -2 flipped the inequality from > to <. That's not optional.

Quadratic Inequalities

This is where most people get lost. You need to find the zeros first, then test regions.

For x² - 5x + 6 > 0:

Step 1: Find zeros by setting the equation equal to zero

x² - 5x + 6 = 0
(x - 2)(x - 3) = 0
x = 2 or x = 3

Step 2: Plot those zeros on a number line. They split into three regions: x < 2, 2 < x < 3, and x > 3.

Step 3: Test each region. Pick a number from each section and plug it into the original inequality.

• x < 2: Try x = 0 → 0² - 5(0) + 6 = 6 > 0 ✓
• 2 < x < 3: Try x = 2.5 → 6.25 - 12.5 + 6 = -0.25 > 0 ✗
• x > 3: Try x = 4 → 16 - 20 + 6 = 2 > 0 ✓

Solution: x < 2 or x > 3. The middle region fails, so you exclude it.

For x² - 5x + 6 < 0, the answer flips: 2 < x < 3 only.

Graphing Zeros

Once you find zeros, plotting them is straightforward. Mark each zero on the x-axis. For linear functions, you have one point. For quadratics, you have one or two. Polynomials can have more.

The zeros tell you where the graph crosses the axis. Between zeros, the graph stays either above or below the axis depending on the function.

For a quadratic with zeros at x = 2 and x = 3, the graph crosses at those points. If the coefficient of x² is positive, the parabola opens upward. If negative, it opens downward.

Connecting Inequalities to Graphs

Here's the visual logic:

• f(x) > 0: You want where the graph sits above the x-axis
• f(x) < 0: You want where the graph sits below the x-axis
• f(x) ≥ 0: Above or on the axis (includes zeros)
• f(x) ≤ 0: Below or on the axis (includes zeros)

The zeros are the boundaries. The inequality sign tells you which side or sides to pick.

Quick Reference: Solving Methods

Getting Started: Step-by-Step Process

For any inequality problem:

1. Set the function equal to zero and solve.
This gives you the critical points where behavior might change.

2. Place those points on a number line.
Mark them clearly. Use open or closed circles based on whether the inequality includes equality.

3. Test one point from each region.
Pick simple numbers. Calculate the result. See if it satisfies the inequality.

4. Write the solution.
Combine all regions that work. Write in interval notation or inequality notation, depending on what your instructor wants.

5. Verify by graphing if possible.
Quick sketch confirms your answer makes sense.

Common Mistakes That Ruin Your Answers

• Forgetting to flip the inequality sign when dividing by negatives
• Testing regions incorrectly by plugging in points near the wrong zero
• Using open circles for ≤ or ≥ signs
• Solving the equation instead of the inequality when that's what you need
• Forgetting that some quadratics have only one zero (the vertex touches once)

Check your sign flips. Check your test points. Those two things fix most errors.

When to Use the Quadratic Formula

Factoring doesn't always work cleanly. If you can't factor a quadratic, use the formula:

x = (-b ± √(b² - 4ac)) / 2a

The discriminant (b² - 4ac) tells you what you're dealing with:

• Positive: Two real zeros
• Zero: One repeated zero (graph touches but doesn't cross)
• Negative: No real zeros (graph stays entirely above or below the axis)

When the discriminant is negative and you have an inequality, the solution is either all real numbers or none. Test any point to find out which.

That's the Core

Zeros are where the function equals zero. Inequalities are where the function is above or below that line. Find the zeros first, then test regions to see which satisfy your inequality. Flip the sign when dividing by negatives. Plot the zeros to visualize what's happening.

Practice with linear inequalities until the sign-flip rule is automatic. Then move to quadratics. The process builds on itself.