Problem 52 - Algebra 2

Question

Which point(s) are solutions of the inequality \(5 x+3 y \geq 2 ?\) 1\. \((0,0)\) II. \((-1,0)\) and \(\left(0,-\frac{2}{3}\right)\) III \(\left(0, \frac{2}{3}\right)\) and \(\left(1,-\frac{2}{3}\right)\) A. I only B. \(|\) and \(\|\) C. III only D. II and III

Step-by-Step Solution

Answer

C. III only

Step 1 - Test point I

Substitute the coordinates of the point (0,0) into the inequality. This gives:5(0) + 3(0) = 0 + 0 = 0 Since 0 is NOT greater than or equal to 2, point I is NOT a solution.

Step 2 - Test points II

Substitute the coordinates of the points (-1,0) and (0, -2/3) into the inequality separately. For (-1,0):5(-1) + 3(0) = -5 + 0 = -5 which is NOT greater than or equal to 2, so (-1,0) is NOT a solution.Now, for (0, -2/3):5(0) + 3(-2/3) = 0 - 2 = -2 which is also NOT greater than or equal to 2, so (0, -2/3) is NOT a solution either.

Step 3 - Test points III

Substitute the coordinates of the points (0, 2/3) and (1, -2/3) into the inequality separately. For (0, 2/3):5(0) + 3(2/3) = 0 + 2 = 2 which is equal to 2, so (0, 2/3) is a solution.Now for (1, -2/3):5(1) + 3(-2/3) = 5 - 2 = 3 which is greater than 2, so (1, -2/3) is also a solution.

Key Concepts

Understanding Algebraic Inequalities

Algebraic inequalities are mathematical expressions that show the relationship between two values, where they are not necessarily equal but rather one is greater or less than the other. These relationships are represented by symbols such as <, >, \(\leq\), or \(\geq\), indicating less than, greater than, less than or equal to, and greater than or equal to, respectively.

In the context of the given exercise, \(5x + 3y \geq 2\) is an algebraic inequality. It indicates that the sum of 5 times the value of \(x\) and 3 times the value of \(y\) should be greater than or equal to 2. To find the solutions to this inequality, specific values for \(x\) and \(y\) can be tested to see if they satisfy the condition given by the inequality.

Key Points to Remember

• An inequality compares two expressions and does not always indicate equality.
• The solution of an inequality is any value or set of values that make the inequality true.
• When you substitute values, pay close attention to whether the result validates the inequality or not.

While working with inequalities, it is crucial to check the solution carefully. If we substitute a value and the inequality holds true, then we have found a valid solution. For example, in step 3 of the solution when (0, 2/3) was substituted, it resulted in a true statement, indicating that this is indeed a solution.

Plotting Points on the Coordinate Plane

The coordinate plane is a two-dimensional surface formed by the intersection of a horizontal number line called the x-axis and a vertical number line called the y-axis. The point where they intersect is known as the origin, or (0,0). Each point on the plane is defined by a pair of numerical coordinates: (x,y) where 'x' is the horizontal position and 'y' is the vertical position.

When visualizing solutions to an inequality like \(5x + 3y \geq 2\), the coordinate plane becomes an invaluable tool. Each point tested in the exercise represents a potential solution plotted on this plane. For instance, point III (0, 2/3) and (1, -2/3) both satisfy the inequality and can be plotted accordingly. Point (0, 0), on the other hand, does not satisfy the inequality and would be plotted outside the solution region.

Visualization Helps

• Graph the inequality to see the complete set of solutions visually.
• Points that lie on the boundary line (when the inequality is an equality) are included in solutions for \(\leq\) or \(\geq\) inequalities.
• Use different markers or colors to distinguish between solutions and non-solutions on the graph.

Visualizing the solutions on a plane can also aid in understanding the concept of a 'solution region' where all points within that region satisfy the inequality.

The Substitution Method for Checking Solutions

The substitution method is a straightforward way to determine whether a particular value or set of values is a solution to an equation or inequality. It involves replacing the variables with the proposed numbers and simplifying the expression to see if the initial statement remains true or false.

In our exercise, the substitution method is used in all three steps to verify potential solutions. For each point, you replace 'x' and 'y' with the respective coordinates and calculate the result. If the inequality holds, then the point is a part of the solution set. For example, when you substituted point III (0, 2/3), the inequality \(5x + 3y \geq 2\) became \(0 + 2 = 2\), which proves the point is a valid solution because the inequality simplifies to \(2 \geq 2\), a true statement.

Benefits of Substitution

• It's a direct and often quick way to check for valid solutions.
• Substitution leaves little room for ambiguity about whether the values satisfy the inequality.
• This method reinforces understanding of how variables interact within an expression.

Remember that while using the substitution method, maintaining proper order of operations and careful arithmetic ensures accurate results. This method not only provides confirmation of solutions but also builds foundational skills for more complex algebraic manipulations.