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Quadratic expressions often appear in algebra, and they can frequently be factored into simpler terms, which makes solving equations or inequalities easier. A quadratic expression typically takes the form: \[ ax^2 + bx + c \]Where \( a \), \( b \), and \( c \) are constants. Factoring quadratics means breaking down this expression into a product of two binomial factors.

In the given problem, we factor the quadratic numerators and denominators to simplify the expression. In the numerator \( x^2 - x - 12 \), we need two numbers that multiply to -12 and add to -1: \(-4\) and \(3\). Thus, the factors are

• \((x - 4)(x + 3)\)

For the denominator \( x^2 + x - 6 \), we look for two numbers that multiply to -6 and add to 1, resulting in the factors:

• \((x + 3)(x - 2)\)

Factoring simplifies the rational expression and is key to identifying its critical points.

Interval notation is a streamlined way to express a range of numbers or solutions on the number line. It provides a clear, concise way to show which parts of the number line are included or excluded in an inequality’s solution.

Intervals can either be open or closed:

• Open intervals: represented by parentheses \( (a, b) \), which means \(a\) and \(b\) are not included in the interval.
• Closed intervals: represented by brackets \( [a, b] \), which means both endpoints \(a\) and \(b\) are included.

For the problem, the solution is expressed as

• \((- ,infty, -3) \cup (-3, 2) \cup (4, \infty)\)

This notation signifies the intervals where the inequality is true, excluding critical points that cause the expression to be undefined or equal to zero.

Solving inequalities involves finding where an expression is greater than, less than, or equal to zero. The original inequality asks where \[\frac{(x - 4)(x + 3)}{(x + 3)(x - 2)} > 0\]Simplifying it means identifying intervals where the expression is positive.

To achieve this:

• Factor both the numerator and the denominator.
• Determine the critical points by setting each factor equal to zero. These points divide the number line into intervals.
• Use test points from each interval to determine the sign of the expression in that interval.

The inequality signals which regions are part of the solution. The correct intervals express where the inequality is satisfied. Solutions are presented without equal signs because the inequality does not include the points that cause zero-value conditions.

Critical points are values that make the polynomial expression zero. They are found by solving each factor in the expression for zero.

Critical points are significant because they can signal potential changes in sign, moving from positive to negative or vice versa. In solving rational inequalities like

• \(\frac{(x - 4)(x + 3)}{(x + 3)(x - 2)} > 0\)

The critical points are:

• \(x = 4\)
• \(x = -3\)
• \(x = 2\)

Once identified, these critical points help establish range intervals in which to test the inequality. Note, however, that not all critical points belong in the solution set, especially if they make the original expression undefined, such as division by zero.