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Quadratic inequalities involve expressions where a quadratic polynomial is less than, less than or equal to, greater than, or greater than or equal to another expression. In the given inequality, \( x^2 \leq |4x - 3| \), the challenge is to interpret and solve the inequality to find the values of \( x \) that satisfy it.
Generally, solving quadratic inequalities involves taking these steps:

• Move all terms to one side to set the expression to zero.
• Factor the quadratic polynomial if possible.
• Find the zeros of the polynomial, which are the solutions when the polynomial equals zero.
• Use these zeros to determine test intervals on the number line.
• Check these intervals to find those that satisfy the inequality.

The test intervals created help identify where the quadratic expression changes sign, which is crucial for determining when the inequality holds true. Make sure to include equality if the inequality sign allows for it, which affects how intervals are noted.

Understanding absolute value is key when dealing with inequalities involving expressions such as \(|4x-3|\). Absolute value refers to the distance a number is from zero on the number line, regardless of direction. For any expression \( |A| \), it can be expressed as:

• \( A \) if \( A \geq 0 \)
• \(-A \) if \( A < 0 \)

This leads to two separate inequalities since we consider both the positive and negative scenarios.
For \(|4x-3|\), this implies:
- The case where \( 4x-3 \geq 0 \): leads to the inequality \( x^2 \leq 4x - 3 \).
- The case where \( 4x-3 < 0 \): leads to \( x^2 \leq -(4x - 3) \).
Solving these separately provides the intervals that satisfy each case, which are then considered collectively.

Interval notation is a way of writing solutions to inequalities in a compact and easy-to-understand form. When used correctly, it shows the range of values that satisfy an inequality. In interval notation:

• Brackets \([\ ]\) denote that the endpoint is included (equal to).
• Parentheses \((\ )\) denote that the endpoint is not included.

In our solution, the intervals \([1, 3]\) and \([-2 - \sqrt{7}, -2 + \sqrt{7}]\) mean that all values between 1 and 3, as well as between \(-2 - \sqrt{7}\) and \(-2 + \sqrt{7}\), are included.
These notations help easily express complex solutions without having to use inequalities with specific larger or smaller comparisons.

The process of solving inequalities involves finding the set of values that make an inequality statement true. This can sometimes be complex, especially when quadratic expressions or absolute values are involved.
First, express the inequality in a standard form. For quadratic expressions, factor them or use the quadratic formula to find roots. These roots will help define the intervals on a number line, serving as boundaries for where the inequality might change from true to false or vice versa.

• Determine the critical points or roots of the expression.
• Test intervals between these points to see where the inequality holds.
• Combine findings from various inequality cases (if needed) to find the total solution set.

In the original problem, this involved two separate inequalities formed from the absolute value brackets, each resulting in distinct solution intervals. These intervals were then combined to address where the original inequality is valid overall.