Problem 1 - Precalculus: Mathematics for Calculus

Question

If the point \((2,3)\) is a solution of an inequality in \(x\) and \(y\) then the inequality is satisfied when we replace \(x\) by ____ and \(y\) by ______ Is the point \((2,3)\) a solution of the inequality \(4 x-2 y \geq 1 ?\)

Step-by-Step Solution

Answer

The point \((2,3)\) satisfies the inequality \(4x - 2y \geq 1\).

Step 1: Understanding the Problem

We need to verify if a given point \((2,3)\) satisfies the inequality \(4x - 2y \geq 1\). This means that when we substitute \(x = 2\) and \(y = 3\) in the inequality, it should result in a true statement if the point is indeed a solution.

Step 2: Substitute the Point

Substitute \(x = 2\) and \(y = 3\) into the inequality \(4x - 2y \geq 1\). This gives us \(4(2) - 2(3) \).

Step 3: Calculate the Left Side

Perform the multiplication: \(4 \times 2 = 8\) and \(-2 \times 3 = -6\). So, the left side of the inequality becomes \(8 - 6\).

Step 4: Evaluate the Expression

Calculate \(8 - 6\), which equals \(2\).

Step 5: Compare with the Right Side

Compare the result \(2\) with the right side of the inequality, which is \(1\). Since \(2 \geq 1\), this inequality is satisfied.

Step 6: Conclusion

Since substituting \((2,3)\) into the inequality results in a true statement, the point \((2,3)\) is indeed a solution to the inequality \(4x - 2y \geq 1\).

Key Concepts

Substitution Method

The substitution method is a powerful tool for solving inequalities, especially when working with variables like \(x\) and \(y\). To apply this method, a specific value is given for each variable in the inequality. By replacing the variables with these values, you can determine whether the given condition holds true or not.

For the exercise provided, the substitution method was used to verify if the point \((2,3)\) satisfied the inequality \(4x - 2y \geq 1\). Here's how it works:

• Start by taking the values provided in the coordinate point. In this case, \(x = 2\) and \(y = 3\).
• Replace \(x\) and \(y\) in the inequality with these values. So, it becomes \(4(2) - 2(3)\).
• Solve the arithmetic expression to see if the inequality is true. If it is, the point satisfies the inequality.

With practice, the substitution method can quickly tell if a specific point is a solution to an inequality.

Coordinate Points

Understanding coordinate points is key when determining solutions for inequalities involving two variables like \(x\) and \(y\). A coordinate point is simply a pair of values representing a position in a two-dimensional space.

In the problem, the point \((2,3)\) is a coordinate point where \(2\) represents the \(x\)-value, and \(3\) represents the \(y\)-value. You use these coordinate points to substitute into inequalities to see if they satisfy the conditions given.

Think of coordinate points as specific locations that tell you where to test or verify conditions in an inequality. Knowing how to properly identify and work with these points helps you solve problems faster and with more accuracy.

• Coordinate points provide clarity in solutions involving graphs or plots.
• They help navigate through graphs, making it easier to find solutions or identify possible errors.

Inequality Solutions

Inequality solutions determine whether a certain condition holds for a range of values or specific points. In our case, we dealt with an inequality in the form \(4x - 2y \geq 1\) and checked whether the point \((2,3)\) was a solution.

To find out if an inequality is satisfied by a point, you perform simple arithmetic operations after substituting the given values. For example, by substituting \(x = 2\) and \(y = 3\) into \(4x - 2y \geq 1\), we calculated that the left side of the inequality equals \(2\), which is indeed greater than \(1\). Thus, the inequality is satisfied.

• Solutions to inequalities can be specific points or ranges of points, depending on the condition.
• Verifying solutions is about ensuring the left-side arithmetic results satisfy the right-side of the inequality.

Evaluating inequality solutions allows us to determine whether points meet the set conditions. In many applications, solving inequalities involves checking multiple points or entire regions, which is useful in real-world problem solving.