Problem 63 - Algebra 2

Question

Determine whether \((0,0)\) satisfies each inequality. Write \(y \leq \frac{3}{4} x-5\)

Step-by-Step Solution

Answer

The point (0,0) does not satisfy the inequality y ≤ (3/4)x - 5.

Step 1: Understand the inequality

The given inequality is \(y \leq \frac{3}{4} x - 5\). This inequality represents a line when it is set to equality (i.e., \(y = \frac{3}{4} x - 5\)), and it describes the region below this line, including the line itself.

Step 2: Insert the coordinates into the inequality

To check if the point \((0,0)\) satisfies the inequality, substitute \(x = 0\) and \(y = 0\) into the inequality. This gives us:\[0 \leq \frac{3}{4}(0) - 5\]

Step 3: Simplify the inequality

Simplify the right side of the inequality:\[0 \leq 0 - 5 = -5\]So, the simplified inequality is \(0 \leq -5\).

Step 4: Evaluate the inequality

Compare the two sides of the simplified inequality. Since \(0\) is not less than or equal to \(-5\), the statement \(0 \leq -5\) is false.

Step 5: Conclusion

Since \(0 \leq -5\) is false, the point \((0,0)\) does not satisfy the inequality \(y \leq \frac{3}{4} x - 5\).

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that allows us to represent geometrical shapes and relationships through algebra using the coordinate plane. A coordinate plane consists of two intersecting perpendicular lines known as axes: the horizontal axis (x-axis) and the vertical axis (y-axis).
Points on the plane are represented as pairs

• A typical point is denoted as (x, y), where x represents the horizontal position and y the vertical position.
• The origin is marked as (0,0), where the x-axis and y-axis intersect.

Coordinate geometry allows us to understand the location of a point or the properties of a shape using numerical equations. We use this system to explore relationships like lines or curves on a graph.

Graphing Inequalities

When we graph inequalities on a coordinate plane, we depict a region fulfilling a specific condition rather than just a line. For the inequality

• For \(y \leq \frac{3}{4}x - 5\), the line \(y = \frac{3}{4}x - 5\) represents the boundary.
• Since the inequality includes the equal sign \(\leq\), the boundary line is solid, indicating points on the line also satisfy the inequality.
• The shading below the line shows all (x, y) points that satisfy the inequality.

To graph, first draw the line as if it were an equation. Then select a test point not on the line, typically the origin (0,0), to decide where to shade. In our example, substituting (0,0) into the inequality shows the region below the line is shaded.

Linear Equations

Linear equations are algebraic expressions involving two variables, usually x and y, which describe a straight line on a coordinate plane. A typical linear equation has the form \(y = mx + b\), where

• \(m\) is the slope, indicating the line's steepness.
• \(b\) is the y-intercept, showing where the line crosses the y-axis.

Linear equations can also be represented as inequalities, such as \(y \leq \frac{3}{4}x - 5\), which not only describes a line but also a range of values. Inequalities like this create half-planes, where one side of the line holds solutions. To determine if a point is a solution, substitute the point's coordinates and check if the inequality holds true.