Problem 9 - Intermediate Algebra

Question

Tell whether the graph of each inequality includes the boundary line. In each case, would the boundary be a solid or a dashed line? a. \(y<3 x-1\) b. \(2 x+3 y \geq-6\) c. \(y \leq-10\) d. \(x>1\)

Step-by-Step Solution

Answer

a. Dashed; b. Solid; c. Solid; d. Dashed.

Step 1: Understand the graphing rules

When graphing linear inequalities, if the inequality sign is "<" or ">", then the boundary line is dashed. If the inequality sign includes "\(\leq\)" or "\(\geq\)", then the boundary line is solid.

Step 2: Identify boundary line for part (a)

For the inequality \(y < 3x - 1\), the symbol is "<", which means the boundary line will be dashed. The line \(y = 3x - 1\) is not part of the solution set.

Step 3: Identify boundary line for part (b)

For the inequality \(2x + 3y \geq -6\), the symbol is "\(\geq\)", so the boundary line will be solid. The line \(2x + 3y = -6\) is included in the solution set.

Step 4: Identify boundary line for part (c)

For the inequality \(y \leq -10\), the symbol is "\(\leq\)", which means the boundary line is solid. The line \(y = -10\) is part of the solution set.

Step 5: Identify boundary line for part (d)

For the inequality \(x > 1\), the symbol is ">", thus the boundary line is dashed. The line \(x = 1\) is not part of the solution set.

Key Concepts

Boundary Line

In graphing inequalities, the boundary line plays a crucial role. It's the line you initially plot on a graph to represent an equation, such as \(y = 3x - 1\) or \(x = 1\). Here's why it's important:

• It serves as the dividing point between the solution region and the non-solution region of an inequality.
• The type of boundary line determined by the inequality symbol tells us whether points on that line are included in the solution set or not.

When graphing, always start by drawing the boundary line as a reference point. The manner in which this line is drawn, solid or dashed, will indicate the nature of the solutions. As you explore inequalities, the boundary line will guide you in shading the appropriate area of the graph that fulfills the inequality condition.

Solid Line

A solid line in the graph of an inequality shows that points on the boundary line are included in the solution set. You will draw a solid line whenever you encounter inequalities with "\(\leq\)" or "\(\geq\)" symbols.Why use a solid line?

• It includes the boundary. All points on this line satisfy the original equation of the boundary line. For example, in the inequality \(y \leq -10\), the line \(y = -10\) includes every point along it in the solution set.
• The solid line demonstrates complete inclusion for any point that lies on the line.

Whenever sketching such inequalities, use a consistent solid line to represent clear inclusion of boundary points in your solution. It contrasts with the dashed line approach, indicating non-inclusion.

Dashed Line

A dashed line is used in graphing inequalities when the solution set does not include the points on the boundary line. You will use a dashed line with inequalities that have symbols like "<" or ">".When to apply a dashed line?

• Markers that indicate exclusions, as they represent that any points residing on the boundary line itself do not satisfy the inequality.
• In cases like \(x > 1\), the dashed line for \(x = 1\) tells us to exclude the line, only considering values of \(x\) greater than 1.

Make sure to draw these lines in a dashed format to avoid confusion. It visually separates potential solutions into those just above or below the line, reaffirming only strict inequalities.