91Ó°ÊÓ

When we talk about the absolute value of a number, we're looking at its distance from zero on a number line, regardless of direction. This means

• For any positive number, the absolute value is the same as the number itself.
• For any negative number, the absolute value is the positive form of that number.

For example, the absolute value of \[ |3| = 3 \] and \[ |-3| = 3 \].
So, when we write \( |x| \), it means:

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

This consideration helps in solving inequalities involving absolute values, like \( |x| \geq x^{2} \). By breaking it down, we can evaluate it separately for positive and negative ranges of \( x \).

Quadratic equations are polynomials of the form \[ ax^2 + bx + c = 0 \], where \( a \), \( b \), and \( c \) are constants. The quadratics are characterized by the square of the variable, \( x^2 \).
To solve quadratic inequalities such as \( x^2 - x \leq 0 \) or \( x(x-1) \leq 0 \), we look for the values of \( x \) that make these expressions true. Quadratic expressions can be factored to find their zeros or roots—these are the points where the expression changes sign.
For instance, to find \( x(x-1) \leq 0 \), set each factor to zero:

• \( x = 0 \)
• \( x = 1 \)

Then examine these intervals: \((-\infty, 0) \), \((0, 1) \), and \((1, \infty) \) by plugging test points into the inequality. Solutions are the intervals where the inequality holds true. This method allows prediction of where curves cross specific axes points.

Graphing inequalities helps to visually identify which parts of the graph lie above or below another curve. Let's look at how to graph \( Y_1 = |x| \) and \( Y_2 = x^2 \):

• \( Y_1 = |x| \) produces a V-shaped graph symmetric about the \( y \)-axis. For \( x \geq 0 \), it equals \( x \), and for \( x < 0 \), it equals \(-x \).
• \( Y_2 = x^2 \) forms a parabolic curve opening upwards with the vertex at (0,0). Its values always increase as \( x \) moves away from the origin.

Plotting these on the same axes allows us to see where \( |x| \) remains above or equal to \( x^2 \), which, as shown in the step-by-step solution, happens in the interval \([-1, 1]\). This visual confirmation further solidifies the calculated solution, enriching the understanding of how inequalities work.