Quadratic Inequalities With Interval Notation

Quadratic inequalities are common in algebra and applied fields like engineering. They involve expressions with variables squared, compared using inequality signs such as $>$, $<$, $\ge$, or $\le$. Solving these inequalities means finding all values of the variable that make the inequality true. Often, the solutions are expressed using interval notation. Interval notation is a clear, compact way to show the range of values that satisfy an inequality. This article explains how to solve quadratic inequalities and write solutions in interval notation effectively.

What Are Quadratic Inequalities?

A quadratic inequality involves a quadratic expression set in relation to zero or another value with inequality signs. For example, x2−3x−4>0x^2 – 3x – 4 > 0x2−3x−4>0 is a quadratic inequality. Here, the goal is to find all $x$ values where the quadratic expression is greater than zero. Unlike quadratic equations, which have specific solutions, quadratic inequalities often have solution sets that are intervals or unions of intervals.

Understanding Interval Notation

Interval notation uses parentheses and brackets to describe continuous sets of numbers. Parentheses $()$ mean the endpoint is not included, while brackets $[]$ mean it is included. For example, $(1,5)$ means all numbers strictly between 1 and 5. The interval $[1,5]$ includes both 1 and 5. When intervals extend infinitely, infinity symbols $\infty$ or $-\infty$ are used, always with parentheses because infinity is not a number that can be included.

Steps to Solve Quadratic Inequalities

1. Rewrite the inequality with zero on one side. For example, write x2−3x−4>0x^2 – 3x – 4 > 0x2−3x−4>0 as is.

2. Find the roots of the related quadratic equation x2−3x−4=0x^2 – 3x – 4 = 0x2−3x−4=0 by factoring or using the quadratic formula.

3. Determine the direction of the parabola by the sign of the coefficient $a$ of x2x^2x2. If $a>0$, it opens upwards; if $a<0$, downwards.

4. Use the roots to split the number line into intervals. These intervals will be tested next.

5. Choose test points from each interval and substitute them back into the inequality.

6. Decide which intervals satisfy the inequality based on the test results.

7. Write the solution in interval notation using brackets or parentheses based on the inequality.

Example 1: Solve x2−3x−4>0x^2 – 3x – 4 > 0x2−3x−4>0

First, solve the quadratic equation x2−3x−4=0x^2 – 3x – 4 = 0x2−3x−4=0. Factor as $(x-4)(x+1)=0$, so roots are $x=4$ and $x=-1$.

Since the leading coefficient $a=1>0$, the parabola opens upwards.

The number line is split into three intervals: $(-\infty ,-1)$, $(-1,4)$, and $(4,\infty )$.

Test points in each interval: pick $x=-2$, $x=0$, and $x=5$.

• For $x=-2$, calculate $4-(-6)-4=4+6-4=6>0$ — true.

• For $x=0$, calculate $0-0-4=-4>0$ — false.

• For $x=5$, calculate $25-15-4=6>0$ — true.

The solution is where the inequality is true: $(-\infty ,-1)\cup (4,\infty )$.

Example 2: Solve −2×2+8x−6≤0 -2x^2 + 8x – 6 \leq 0−2x2+8x−6≤0

Rewrite the inequality and solve −2×2+8x−6=0 -2x^2 + 8x – 6 = 0−2x2+8x−6=0.

Divide both sides by -2 to simplify: x2−4x+3=0x^2 – 4x + 3 = 0x2−4x+3=0.

Factor: $(x-3)(x-1)=0$, roots are $x=1$ and $x=3$.

Since the original parabola opens downward ($a=-2<0$), the shape is inverted.

Intervals: $(-\infty ,1)$, $(1,3)$, and $(3,\infty )$.

Test points: $x=0$, $x=2$, and $x=4$.

• For $x=0$, substitute in original: $0+0-6=-6\le 0$ — true.

• For $x=2$, substitute: $-8+16-6=2\le 0$ — false.

• For $x=4$, substitute: $-32+32-6=-6\le 0$ — true.

Because the inequality is $\le 0$, include roots with brackets.

Solution is $(-\infty ,1]\cup [3,\infty )$.

Why Use Interval Notation?

Interval notation is concise and clear. It neatly shows ranges without repeating inequality symbols. It works well in calculus, algebra, and applied math. Interval notation helps when expressing solution sets on graphs or when communicating solutions quickly and unambiguously.

Conclusion

Quadratic inequalities with interval notation provide an effective way to express solution ranges. By solving the related quadratic equation, testing intervals, and writing answers in interval notation, you can clearly show where inequalities hold true. This approach simplifies understanding and applying quadratic inequalities in math and real-world problems.