Problem 62 - Algebra 2

Question

Determine whether \((0,0)\) satisfies each inequality. Write \(y \geq-x+1\)

Step-by-Step Solution

Answer

No, \((0,0)\) does not satisfy the inequality \(y \geq -x + 1\).

Step 1: Identify the inequality

Given inequality: \( y \geq -x + 1 \). We need to check if the point \((0,0)\) satisfies this inequality.

Step 2: Substitute the point into the inequality

Substitute \(x = 0\) and \(y = 0\) into the inequality. The inequality becomes \(0 \geq -0 + 1\).

Step 3: Simplify the inequality

Simplify the right side of the inequality: \(0 \geq 1\).

Step 4: Evaluate the inequality

The simplified inequality \(0 \geq 1\) is not true, because 0 is not greater than or equal to 1.

Key Concepts

Substitution Method

The substitution method is a mathematical technique used to determine whether a specific point lies on a graph defined by an equation or inequality. In the context of inequalities, this method involves substituting the coordinates of a point into the inequality to check whether the inequality holds true.
You simply replace each variable in the inequality with the respective coordinate values from the point. For example, if you have the inequality \( y \geq -x + 1 \) and the point \((0,0)\), you substitute \(x = 0\) and \(y = 0\) into the inequality, making it \(0 \geq -0 + 1\). This reformation allows you to assess the truth of the inequality at the given point.
By using substitution this way, you can effectively analyze the inequality without needing to graph it, providing a straightforward approach to determining whether a point satisfies an inequality.

Inequality Evaluation

Inequality evaluation is the process of determining whether a mathematical statement involving inequality signs like \(\geq\), \(\leq\), \(>\), or \(<\) is true or false. It goes beyond simple arithmetic by dealing with relationships and comparing values.
For the inequality \(0 \geq 1\), derived from substituting into the original inequality, you need to compare the left-hand side (0) with the right-hand side (1). Here, the statement 0 is greater than or equal to 1 is false because 0 is neither greater than nor equal to 1.
Inequality evaluation largely hinges on understanding and interpreting these symbols correctly. It's crucial for solving problems in algebra, enabling you to draw conclusions about the possible values and relationships in mathematical and real-world contexts.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is a branch of mathematics that uses coordinates to describe geometric figures and solve problems. It combines algebra and geometry to provide a connection between numerical coordinates and geometric structures.
In coordinate geometry, inequalities like \( y \geq -x + 1 \) define regions on a coordinate plane. These inequalities can represent lines, and the solutions are often areas above or below these lines depending on the inequality. For example, \( y \geq -x + 1 \) represents all the points lying on or above the line \(y = -x + 1\).
When tasked with determining whether a particular point, such as \((0,0)\), satisfies the inequality, coordinate geometry allows for direct visualization of where the point lies in relation to the line and shaded region defined by the inequality. This visualization aids in verifying if the coordinates meet the required conditions of the inequality.