Problem 43 - Algebra A Combined Function

Question

Determine whether (1,1) is included in each graph. $$ 3 x+4 y<8 $$

Step-by-Step Solution

Answer

(1,1) is included in the graph of the inequality as it satisfies \(3x + 4y < 8\).

Step 1: Substitute the Point in the Inequality

First, let's substitute the point \((1, 1)\) into the inequality \(3x + 4y < 8\). We replace \(x\) with \(1\) and \(y\) with \(1\), resulting in the expression \(3(1) + 4(1) < 8\).

Step 2: Simplify the Expression

Now, simplify the left side of the expression: \(3(1) + 4(1) = 3 + 4 = 7\). Substitute back into the inequality to get \(7 < 8\).

Step 3: Interpret the Result

The inequality \(7 < 8\) is true, which means that the point \((1, 1)\) satisfies the inequality \(3x + 4y < 8\).

Key Concepts

Understanding Inequalities

Inequalities are mathematical statements used to compare two values or expressions. They show that one side is less than, greater than, less than or equal to, or greater than or equal to the other side. In this exercise, we deal with a strictly less than inequality: \(3x + 4y < 8\). This means we are looking for all pairs of \((x, y)\) that make the left side smaller than 8.

Here's how it works:

• The inequality \(3x + 4y < 8\) defines a region in a coordinate plane instead of just a line.
• Every solution is a point \((x, y)\) in this defined region, illustrating an infinite set of possibilities rather than a fixed point.

Graphing inequalities involves shading this region, allowing us to visualize all solutions at once. The line created by \(3x + 4y = 8\) acts as a boundary, but it is not part of the solution set since we are dealing with a strict inequality (<). Points exactly on this line do not satisfy the inequality.

Exploring the Coordinate Plane

The coordinate plane is a two-dimensional space that helps us visualize equations and inequalities. It consists of horizontal and vertical lines called axes that intersect at a point called the origin, denoted as \((0, 0)\). In analyzing the inequality \(3x + 4y < 8\), the coordinate plane provides a platform to test if points like \((1, 1)\) satisfy the inequality.

Here's a breakdown of its components:

• The horizontal line is the x-axis, and the vertical line is the y-axis.
• Any point on the plane is expressed as \((x, y)\), where \(x\) is the ordinate and \(y\) is the abscissa.
• The graph of the line \(3x + 4y = 8\) divides the plane into regions, helping determine where \(3x + 4y < 8\) holds true.

Using the coordinate plane enables us to graph inequalities by shading the region of solutions. In this exercise, knowing whether the point \((1, 1)\) falls within this region helps verify if it is part of the solution set.

Applying the Substitution Method

The substitution method is a technique used in solving equations and inequalities by replacing variables with specific values. In this exercise, we use substitution to check point \((1, 1)\) against the inequality \(3x + 4y < 8\).

Here's a step-by-step explanation of the process:

• Take the given inequality, \(3x + 4y < 8\).
• Substitute the value of \(x\) with 1 and \(y\) with 1, resulting in \(3(1) + 4(1) < 8\).
• Simplify the inequality to find \(7 < 8\).
• Since the expression \(7 < 8\) is true, the point \((1, 1)\) satisfies the inequality.

By using substitution, we can directly verify if a specific point lies within the solution set of the inequality. It's a straightforward approach to confirm individual solutions by testing given points against the inequality.