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Quadratic inequalities are inequalities that involve a quadratic expression, which is an expression containing a term with a square, like \(x^2\). This type of inequality can often be expressed in the form \(ax^2 + bx + c \leq 0\), \(ax^2 + bx + c \geq 0\), or with strict inequalities like \(<\) or \(>\). Solving quadratic inequalities involves finding the values of \(x\) for which the inequality holds true. The solution process typically involves the following steps:

• Moving all terms to one side to create an expression set less than or greater than zero.
• Factoring the quadratic expression to find its roots, also known as critical points.
• Testing intervals determined by the critical points to see where the inequality is satisfied.
• Using these intervals to express the solution in interval notation, which provides a concise way of describing ranges of values for \(x\).

As an example, consider the inequality \(x^2 - 2x + 8 \leq 2x + 5\). By rearranging it and then factoring, you can find the critical points, leading to interval testing.

Factoring is a crucial method used to solve quadratic inequalities and equations. A quadratic expression of the form \(x^2 + bx + c\) can frequently be rewritten as a product of two binomials, namely \((x + m)(x + n)\), where:

• \(m\) and \(n\) are numbers whose product is \(c\) and sum is \(b\).

Finding these numbers involves identifying pairs of factors of \(c\) that add up to \(b\). For example, in the expression \(x^2 - 4x + 3\), the product of the two numbers should be 3 and their sum should be -4. Factors that fulfill these conditions are -1 and -3, leading to the factorization:\[ (x - 1)(x - 3) \]This factorization helps us identify the critical points or zeros of the expression, which are essential when examining the intervals on the number line.

Interval notation is a succinct way to describe the set of solutions to an inequality. It uses brackets and parentheses to denote intervals and whether endpoints are included in the solution:- **Square brackets** \([\), \(]\) indicate that an endpoint is included (\(\leq\) or \(\geq\)).- **Parentheses** \((\), \()\) show that an endpoint is not included (\(<\) or \(>\)).For example, if after checking the intervals in a quadratic inequality, you find that values between 1 and 3 satisfy the condition, with both endpoints included, the interval notation is \([1, 3]\). This notation means all \(x\) in the range from 1 to 3, inclusive, satisfy the inequality.

Critical points in a quadratic inequality are the values of \(x\) where the quadratic expression equals zero. These points are found by solving \((x - m)(x - n) = 0\), resulting in \(x = m\) and \(x = n\). In the inequality \((x - 1)(x - 3) \leq 0\), the critical points are 1 and 3. Graphically, these points are where the parabola intersects the x-axis.Determining critical points helps divide the number line into intervals that can be tested to see where the inequality holds. After finding critical points, test intervals:

• Choose test values from the intervals formed around critical points.
• Check the signs of the expression at these points.
• Determine which intervals satisfy the inequality.

By including critical points in interval notation if they satisfy the inequality, you can accurately describe the set of \(x\) values that meet the required conditions.