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Absolute value is a concept that measures the distance of a number from zero on a number line, regardless of direction. It's often denoted as \( |a| \), where \( a \) is any real number. The key property of absolute value is that it turns any negative number into a positive version of itself.
- **Positive Numbers**: For \( a > 0 \), \( |a| = a \). - **Negative Numbers**: For \( a < 0 \), \( |a| = -a \). - **Zero**: For \( a = 0 \), \( |a| = 0 \).
Let's apply this to inequalities. Consider the expression \( |x-y| < 3 \). This states that the absolute difference between \( x \) and \( y \) is strictly less than 3. We write it as two simultaneous inequalities: \( x-y < 3 \) and \( x-y > -3 \). In coordinate geometry, this represents a band or strip of points between two parallel lines. The understanding of absolute value helps in defining such regions on the coordinate plane. It guides you to determine how far points can be from each other while still satisfying the given inequality.

Graphing inequalities on the coordinate plane involves shading the region that satisfies an inequality. We start graphing points but extend that to regions, giving a visual interpretation of all possible solutions.
To handle inequalities like \( |x-y+5| < 3 \), we begin by expressing the absolute value as a compound inequality: \( -3 < x-y+5 < 3 \). This splits into two linear inequalities: \( x-y < -2 \) and \( x-y > -8 \). Here, graphing involves plotting the boundary lines \( x-y=-2 \) and \( x-y=-8 \), which are not included in the solution (hence dashed lines).
- **Shade the Correct Region**: Once the lines are drawn, shade the area between them to illustrate the solution. - **Boundary Lines**: Always determine if lines are included in the solution using dashed or solid lines.When graphing, it helps to test a point not on the boundary to see if it lies in the solution region. Graphing inequalities turns abstract expressions into tangible visual realities, facilitating deeper understanding of their solutions.

Coordinate geometry, or analytic geometry, combines algebra and geometry. It allows us to study geometry using a coordinate plane, giving a precise method for plotting points and lines.
The basic idea involves plotting points with coordinates \( (x, y) \). Lines on the plane can be expressed as equations, like \( x-y=3 \), which rearranges to describe a line with a certain slope and intercept. For example, lines such as \( x-y=3 \) or \( x-y=-3 \) have the same slope but different intercepts. This creates parallel lines.
- **Intersections and Regions**: When solving inequalities, these equations define boundaries. The region of solutions is often between or outside these boundary lines.- **Slopes and Intercepts**: Understanding the slope (rise over run) and intercepts (where the line crosses axes) is crucial for plotting accurate lines.In problems like \( |x-y| < |x+y| \), coordinate geometry helps to visualize and understand which regions satisfy the condition. You're visually separating the plane into regions that meet the inequality's criteria, significantly easing the solution process.