Quadratic Inequalities

Welcome, students! 😊 In this lesson, you will learn how to solve and interpret quadratic inequalities—statements like $x^2-5x+6>0$ or $2x^2+3x-2\le 0$. These are important in IB Mathematics Analysis and Approaches SL because they connect algebra, graphing, and real-world decision-making. By the end of this lesson, you should be able to identify the parts of a quadratic inequality, solve it using graphs or algebra, and explain the solution in context.

Lesson objectives:

• Understand the language and meaning of a quadratic inequality.
• Solve quadratic inequalities using factorisation, graphs, and sign analysis.
• Interpret solutions as intervals on the number line.
• Connect quadratic inequalities to quadratic functions and their graphs.
• Use examples to show how these ideas appear in real situations.

1. What is a quadratic inequality?

A quadratic inequality compares a quadratic expression to another value using symbols such as $<$, $>$, $\le$, or $\ge$. A quadratic expression has the form $ax^2+bx+c$, where $a\ne 0$. A quadratic inequality might look like $ax^2+bx+c>0$, $ax^2+bx+c\le 4$, or $x^2-9<0$.

The key idea is that you are not finding one exact answer only. Instead, you are finding all values of $x$ that make the inequality true. That answer is often a set of intervals. For example, the solution to $x^2-4>0$ is all $x$ values less than $-2$ or greater than $2$.

This connects directly to functions because a quadratic expression can be written as a function, such as $f(x)=x^2-4$. Then the inequality $x^2-4>0$ becomes $f(x)>0$. You are asking where the graph of $f(x)$ lies above the $x$-axis. That graph-based thinking is a major IB skill 📈.

A quadratic graph is a parabola. If the parabola opens upward, values of $f(x)$ are positive outside the roots and negative between them, depending on the position of the graph. If the parabola opens downward, the sign pattern is reversed. This sign behavior is the heart of quadratic inequalities.

2. Solving by factorising and using a sign diagram

One of the most useful methods is to solve the related quadratic equation first. Start with the inequality, then change it temporarily into an equation to find the critical values or roots. These are the points where the expression equals zero.

Example: solve $x^2-5x+6\ge 0$.

First, factorise the quadratic:

$$x^2-5x+6=(x-2)(x-3)$$

Now find where each factor is zero: $x=2$ and $x=3$. These divide the number line into three intervals:

• $x<2$
• $2<x<3$
• $x>3$

Test one value from each interval:

• For $x=0$, $(0-2)(0-3)=6$, which is positive.
• For $x=2.5$, $(2.5-2)(2.5-3)$ is negative.
• For $x=4$, $(4-2)(4-3)=2$, which is positive.

So the expression is non-negative outside the roots and negative between them. Because the inequality is $\ge 0$, include the roots too. The solution is

$$x\le 2 \quad \text{or} \quad x\ge 3$$

This method is reliable because it shows where the expression changes sign. The roots split the number line into regions, and each region keeps the same sign unless a root is crossed.

A sign diagram helps you organize this clearly. For a quadratic with two distinct real roots and positive leading coefficient $a>0$, the sign pattern is usually positive, negative, positive across the intervals. If $a<0$, it is negative, positive, negative.

3. Using the graph of a quadratic function

Another method is to use the graph of the quadratic function. If $f(x)=ax^2+bx+c$, then solving an inequality like $f(x)>0$ means finding where the graph is above the $x$-axis. Solving $f(x)<0$ means finding where it is below the $x$-axis.

Example: consider $f(x)=-(x-1)(x+2)$.

The roots are $x=1$ and $x=-2$. Since the coefficient of $x^2$ is negative, the parabola opens downward. That means the graph is above the $x$-axis between the roots and below the $x$-axis outside them.

So:

• $f(x)>0$ for $-2<x<1$
• $f(x)\le 0$ for $x\le -2$ or $x\ge 1$

This is why understanding transformations matters. If you know how the graph moves, you can predict the inequality’s solution set. For example, if $y=x^2$ shifts up by 3 units to become $y=x^2+3$, then $x^2+3>0$ is true for all real numbers because the graph never touches or crosses the $x$-axis.

Graphing technology can also help check answers, but IB expects you to explain the reasoning, not just read the final interval from a screen. Always connect the graph to the inequality symbol.

4. Special cases and interval notation

Not every quadratic inequality has two real roots. This matters a lot.

Case 1: Two real roots

Example: $x^2-1<0$ factors as $(x-1)(x+1)<0$. The roots are $-1$ and $1$. Since the parabola opens upward, the expression is negative between the roots, so the answer is

$$-1<x<1$$

Case 2: One repeated root

Example: $(x-2)^2\ge 0$.

A square is always non-negative, so the inequality is true for every real number. The solution is

$$x\in\mathbb{R}$$

If the inequality were $(x-2)^2<0$, there would be no solution, because a square cannot be negative.

Case 3: No real roots

Example: $x^2+4>0$.

The graph of $y=x^2+4$ is always above the $x$-axis, so the inequality is true for all real numbers. However, $x^2+4<0$ has no real solution.

When writing final answers, use interval notation or inequality notation clearly. For example:

• $(-\infty,-2]\cup[3,\infty)$
• $-2\le x\le 3$
• $x\in\mathbb{R}$
• no solution

In IB, precision matters. If the inequality includes equality, such as $\le$ or $\ge$, then the roots are included in the answer. If it uses $<$ or $>$, then the roots are excluded.

5. Real-world meaning and IB-style interpretation

Quadratic inequalities often describe limits, safety ranges, or conditions in context. Suppose the height of a ball is modeled by $h(t)=-5t^2+20t+1$, where $t$ is time in seconds. If you want to know when the ball is at least $6$ meters high, solve

$$-5t^2+20t+1\ge 6$$

Rearrange to get

$$-5t^2+20t-5\ge 0$$

Then divide by $-5$ and remember to reverse the inequality sign:

$$t^2-4t+1\le 0$$

Now find the roots using the quadratic formula if needed:

$$t=\frac{4\pm\sqrt{16-4}}{2}=2\pm\sqrt{3}$$

Because the parabola opens upward and the inequality is $\le 0$, the solution is between the roots:

$$2-\sqrt{3}\le t\le 2+\sqrt{3}$$

This means the ball is at least $6$ meters high during that time interval.

This kind of question shows why quadratic inequalities are useful in the real world. They describe when a quantity stays within a target range. In mathematics, the graph gives the structure, and the inequality symbol tells you which regions matter.

Conclusion

Quadratic inequalities combine algebra and function thinking. students, you now know that the solution comes from identifying the roots of the related quadratic equation, then using the graph or sign pattern to decide where the expression is positive or negative. You also know that the answer is usually a set of intervals, not a single number. This topic is an important part of IB Mathematics Analysis and Approaches SL because it builds fluency with graphs, algebraic reasoning, and interpretation in context. Remember: solve the related equation, check the sign of the parabola, and match your final interval to the inequality symbol ✅.

Study Notes

• A quadratic inequality has the form $ax^2+bx+c<0$, $ax^2+bx+c\le 0$, $ax^2+bx+c>0$, or $ax^2+bx+c\ge 0$.
• First solve the related equation $ax^2+bx+c=0$ to find the roots.
• The roots split the number line into intervals where the expression keeps a constant sign.
• If $a>0$, the parabola opens upward; if $a<0$, it opens downward.
• Use factorisation, sign diagrams, or graphs to determine where the inequality is true.
• If the inequality includes $\le$ or $\ge$, include the roots in the solution.
• If the inequality uses $<$ or $>$, do not include the roots.
• A quadratic inequality can often be interpreted as where a function is above or below the $x$-axis.
• Final answers should be written clearly using interval notation, inequality notation, or both.
• In context, quadratic inequalities describe ranges of values that satisfy a condition, such as time, height, cost, or area.