Problem 29 - Algebra A Combined Function

Question

Graph each inequality. $$ 2 x+7 y>5 $$

Step-by-Step Solution

Answer

Shade above the dashed line of \(2x + 7y = 5\).

Step 1: Rewrite the Inequality as an Equation

To graph the inequality, first rewrite it as an equality to find the corresponding line. Change \( 2x + 7y > 5 \) to \( 2x + 7y = 5 \).

Step 2: Find the Intercepts

Solve for the intercepts by setting first \( x = 0 \) and then \( y = 0 \). For \( x = 0 \), \( 7y = 5 \implies y = \frac{5}{7} \). For \( y = 0 \), \( 2x = 5 \implies x = \frac{5}{2} \). The intercepts are \((0, \frac{5}{7})\) and \((\frac{5}{2}, 0)\).

Step 3: Draw the Line

Plot the points \((0, \frac{5}{7})\) and \((\frac{5}{2}, 0)\) on a graph. Draw a dashed line through these points since the inequality is 'greater than' (strict inequality), indicating that points on the line are not included in the solution.

Step 4: Determine the Half-Plane

Select a test point not on the line, such as \((0,0)\). Substitute it into the inequality: \( 2(0) + 7(0) = 0 \). Since \( 0 ot> 5 \), the region that does not include \((0,0)\) (the region above the line) is the solution.

Step 5: Shade the Solution Region

Shade the region above the dashed line, where all points satisfy the inequality \( 2x + 7y > 5 \). This shaded half-plane represents all solutions to the inequality.

Key Concepts

Linear Inequalities

Linear inequalities, like the one given as \( 2x + 7y > 5 \), describe a relationship between two variables, \( x \) and \( y \). These inequalities differ from linear equations because they indicate a region on a graph where solutions exist, rather than a single line. The inequality symbol \( > \) suggests that the region of solutions contains points where the expression on the left is greater than that on the right.
When graphing linear inequalities, the first step often involves rewriting the inequality as an equation. By doing this, we can visualize the boundary line that separates the solutions from non-solutions. For our example, this means converting \( 2x + 7y > 5 \) to the equation \( 2x + 7y = 5 \).
It's essential to remember that **strict inequalities** like \( > \) or \( < \) indicate that the line itself is not part of the solution. This is why we use a dashed line to represent the equation on the graph.

Coordinate Plane

The coordinate plane is integral to graphing linear inequalities and understanding solutions visually. It consists of two number lines intersecting at a right angle: the horizontal number line (x-axis) and the vertical number line (y-axis). They divide the plane into four quadrants. Each point on the plane is an ordered pair \((x, y)\) denoting the horizontal and vertical positions, respectively.
For graphing, knowing how to plot points accurately is vital. The intercepts \((0, \frac{5}{7})\) and \((\frac{5}{2}, 0)\) give us specific locations to start drawing our line. Placing these points correctly ensures the boundary line reflects the equation accurately. The notion of dividing the plane into regions through lines is fundamental in expressing inequalities geometrically.

Solution Region

The solution region of a linear inequality refers to the area on the graph containing all points that satisfy the inequality. Once we have our boundary line from the inequation's corresponding equation, our task is to determine which area represents the solutions.
The dashed line in our graph divides the coordinate plane into two regions or half-planes. Since our inequality is \( 2x + 7y > 5 \), the actual solution is the region above this line. This is because, for each point in this region, a substitute into the inequality yields a true statement, satisfying the condition \( 2x + 7y \) is indeed greater than \(5\).
Shading helps indicate clearly which part of the graph pertains to the solution set. This visual mark makes it simple to understand precisely which side of the boundary line forms the solution.

Test Points

Choosing a test point is a clever technique to identify the solution region of a linear inequality. Test points are any chosen points on the graph not lying on the boundary line. A common choice is the origin point \((0,0)\) because calculations are usually easier.
Upon substituting this test point into the inequality \( 2x + 7y > 5 \), we find: \( 2(0) + 7(0) = 0 \). This statement, \( 0 > 5 \), is false, indicating that \((0, 0)\) is not in the solution set. Therefore, the opposite region, not containing the test point, is the solution area. This method helps avoid potential errors and confirms the correctness of the solution region.
Other points can also serve as test points, and using them provides extra assurance of the shaded region's accuracy on the graph.