Problem 61 - Algebra 2

Question

Determine whether \((0,0)\) satisfies each inequality. Write \(y<2 x+3\)

Step-by-Step Solution

Answer

Yes, (0,0) satisfies the inequality.

Step 1: Substitute the values

To determine if the point \((0,0)\) satisfies the inequality, substitute \(x=0\) and \(y=0\) into the inequality \(y < 2x + 3\).

Step 2: Simplify the inequality

Substitute \(x=0\) and \(y=0\) into the inequality and simplify: \(0 < 2(0) + 3\).

Step 3: Calculate the expression

Calculate the expression on the right-hand side: \(2(0) + 3 = 0 + 3 = 3\).

Step 4: Determine the inequality result

The inequality becomes \(0 < 3\), which is true.

Key Concepts

Substitution Method

The substitution method is a valuable tool to verify if a given point satisfies an inequality. It involves replacing variables in an equation or inequality with specific values to see if the statement holds true. In the context of the inequality \( y < 2x + 3 \), we substitute \( x = 0 \) and \( y = 0 \) to check if the point \((0, 0)\) is a solution.

This process involves plugging in the values directly into the inequality:

• Substitute \(x = 0\) and \(y = 0\) into the inequality \( y < 2x + 3 \).
• Simplify the expression as \( 0 < 2(0) + 3 \).
• This results in \(0 < 3\).

By following this method, we establish that the inequality holds true for the substituted values, meaning the point \((0, 0)\) does indeed satisfy the inequality.

Coordinate Geometry

Coordinate geometry provides a graphical context to understand inequalities. It involves a plane where each point is defined by coordinates, commonly \(x\) and \(y\). In inequalities like \(y < 2x + 3\), coordinate geometry helps visualize the solution set.

The inequality \(y < 2x + 3\) represents a region below the line \(y = 2x + 3\) on a graph:

• The line \(y = 2x + 3\) is the boundary but not part of the solution set.
• Any point below this line would satisfy the inequality.
• The point \((0, 0)\) is under the boundary line, illustrating it fulfills the condition.

Coordinate geometry allows us to draw not just lines, but shapes of solution areas, offering a visual understanding of linear inequalities.

Linear Inequalities

Linear inequalities, unlike linear equations, describe a relationship using symbols like \(<, >, \leq,\) and \(\geq\). These inequalities indicate a range of possible solutions rather than a single point.

Understanding linear inequalities:

• They describe half-planes when graphed on a coordinate plane.
• The boundary line, such as \(y = 2x + 3\), helps ascertain which half-plane is a solution.
• Inequalities like \(y < 2x + 3\) use dashed lines to indicate that points on the line are not included.

This conceptual framework helps us interpret solutions of inequalities algebraically and visually, showing the set of all possible solutions that satisfy the given conditions.