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When we talk about **linear inequalities**, we are referring to expressions that involve an inequality sign (such as ">", "<", "≥", or "≤") with linear terms. A linear term is simply something like "5x" or "3y", where the variables have a power of one without higher exponents like squares or cubes.
Linear inequalities can describe a portion of the number line or the coordinate plane where the inequality holds true. Just like equations, linear inequalities can be rearranged and simplified using algebraic principles.

• Example: For the inequality \(3x + 4 \leq 10\), you can subtract 4 from both sides to get \(3x \leq 6\), and then divide by 3 to get \(x \leq 2\).

The solutions to linear inequalities can be graphed on a number line or a two-dimensional plane, which leads us to our next topic.

**Coordinate geometry** involves plotting points, lines, and curves on a coordinate plane. It helps visualize mathematical concepts by placing them onto a set grid with an x-axis and a y-axis. Each point on this grid is identified by a pair of numbers known as coordinates.
Understanding the coordinate plane is fundamental for graphing equations and inequalities. Here are some basic elements:

• The x-axis is the horizontal line, while the y-axis is the vertical line. Together, they form the plane.
• Coordinates are written as (x, y), where x represents the horizontal position, and y the vertical.

For inequalities like \(5x + 3y \geq -15\), coordinate geometry helps us understand where this inequality holds true. When an inequality is graphed, it usually covers an area of the plane rather than just a line.

**Graphing lines** is an essential skill in algebra and geometry. A line in the coordinate plane is often described by an equation like \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.
The steps to graphing a line include selecting specific points that satisfy the equation, such as intercepts:

• X-intercept: The point where the line crosses the x-axis, found by setting \(y = 0\).
• Y-intercept: The point where the line crosses the y-axis, found by setting \(x = 0\).

Once you have these points, use a ruler to connect them, creating a line. For inequalities like \(5x + 3y \geq -15\), the line you draw is part of the graph but not the entire solution.
To determine which side of the line to shade, a test point like (0,0) is used. If the test point satisfies the inequality, that entire side is shaded, indicating all the solutions to the inequality.