Problem 64 - Algebra 2

Question

Determine whether \((0,0)\) satisfies each inequality. Write \(2 x+6 y+3>0\)

Step-by-Step Solution

Answer

The point \((0, 0)\) satisfies the inequality \(2x + 6y + 3 > 0\).

Step 1: Understand the Inequality

We need to determine if the point \((0, 0)\) satisfies the inequality \(2x + 6y + 3 > 0\). This means we're checking if, after substituting \(x = 0\) and \(y = 0\) into the inequality, the expression is still greater than zero.

Step 2: Substitute Values

Substitute \(x = 0\) and \(y = 0\) into the inequality:\[2(0) + 6(0) + 3 > 0.\]

Step 3: Simplify the Expression

Simplifying the expression, we get:\[0 + 0 + 3 > 0,\] which simplifies to \(3 > 0\).

Step 4: Determine the Truth Value

Since \(3 > 0\) is a true statement, the point \((0, 0)\) does indeed satisfy the inequality.

Key Concepts

Substitution Method

The substitution method is a straightforward technique commonly used to solve mathematical equations and inequalities. When faced with an inequality or equation, this method involves replacing the variables with given numbers to check if a particular outcome is true.

In our exercise, we were given the inequality \(2x + 6y + 3 > 0\) and asked to check whether the point \((0, 0)\) satisfies it. To do this using substitution:

• We replaced \(x\) with 0.
• We also replaced \(y\) with 0.

This simplifies the inequality to \(2(0) + 6(0) + 3 > 0\), which is straightforward to evaluate further. By following this approach, we simplify the problem to basic arithmetic, allowing us to easily determine whether the initial inequality holds true under the given conditions.

System of Inequalities

A system of inequalities involves multiple inequalities that are considered together. These inequalities can be solved simultaneously to find a range of solutions that satisfy all conditions set by the equations.

Although our exercise only involved one inequality, understanding a system of inequalities is essential. Imagine two or more inequalities with shared variables. To find solutions, you'd:

• Substitute values if specific points are given, just like with individual inequalities.
• Consider potential intersections of solutions provided by graphed inequalities in a system.

This approach allows us to identify feasible solutions that meet all given constraints, which is crucial in many real-world scenarios like economics, engineering, and optimization problems.

Graphing Inequalities

Graphing inequalities is a visual way to find and understand solutions for inequalities. By plotting the inequality on a coordinate plane, you can see the regions that represent solutions.

In our exercise, to visually determine if \((0,0)\) satisfies an inequality like \(2x + 6y + 3 > 0\), you'd follow these steps:

• First, treat the inequality as an equation (e.g., \(2x + 6y + 3 = 0\)) to find the boundary.
• Then, draw the line on a graph that represents this boundary.
• Shade the area above or below the line based on the inequality sign (above for \(>\), below for \(<\)).

For the inequality \(2x + 6y + 3 > 0\), you would shade above the line. The region containing the origin \((0,0)\) indicates whether it satisfies the inequality. Since it does, the point \((0,0)\) is part of the shaded solution area.