Problem 13 - Thinking Mathematically

Question

Graph each linear inequality. \(y \leq 3 x+2\)

Step-by-Step Solution

Answer

The graph of \(y \leq 3x + 2\), includes a solid line along \(y = 3x + 2\), and the area below this line is shaded to show all the solutions.

Step 1: Understand the inequality

The linear inequality \(y \leq 3x + 2\) has slope 3 and y-intercept is 2. Also, because it's a 'less than or equal to' inequality, the graphed line will be solid and solutions lie on the line and below it.

Step 2: Graph the line

Start by plotting the y-intercept (2) on the y-axis, then use the slope (3) to find another point. Since the slope is 3 (can be written as 3/1), starting at y-intercept, go up 3 units and right 1 unit to find another point (1, 5). Draw a solid line through these points, since it's a 'less than or equal to' inequality.

Step 3: Shade the solution area

The inequality is \(y \leq 3x + 2\), which means that the possible values for y are located on the line and below it. So, shade the half-plane below the line to represent all the solutions to the inequality.

Key Concepts

Understanding Linear Inequality

Linear inequalities are quite similar to linear equations but with an important twist — instead of an equal sign (=), they involve inequality signs like <, >, ≤, or ≥. When you see an inequality like \(y \leq 3x + 2\), it tells you that y is less than or equal to the value of 3x + 2 for any given x. Unlike an equation, which has only one line as a solution, an inequality indicates a range of possible solutions forming an area on the graph.

Imagine you're plotting the boundaries of a park where y represents the southern limit, and 3x + 2 is the northern boundary. The inequality \(y \leq 3x + 2\) means every point in the park (the solution area) must be at or below this northern boundary.

Slope-Intercept Form

To graph a linear inequality, we first need to understand the slope-intercept form of a linear equation, which is y = mx + b. Here, m represents the slope of the line, and b is the y-intercept, the point where the line crosses the y-axis.

In our inequality \(y \leq 3x + 2\), the slope is 3 and the y-intercept is 2. Picture the slope as a staircase that you can ascend or descend — in this case, for every step to the right along the x-axis (1 unit), you'll climb 3 steps up in the y-direction. To graph this line, start at the y-intercept (0,2) and use the slope to find another point by moving up 3 units and to the right 1 unit, reaching the point (1,5). Finally, connect these dots with a solid line to form the northern boundary of our park (representing our inequality).

Shading the Solution Area

Now to visualize the full set of solutions for the inequality \(y \leq 3x + 2\), we need to shade the correct side of the boundary line. Since the inequality shows that y is less than or equal to 3x + 2, the area representing possible solutions will be at or below the line we've drawn.

Shading is like painting the park's available area — you want to make clear which side of the boundary visitors are allowed to enjoy. For our example, you'd shade downward from the line towards the negative y-direction. This shaded region of the graph represents all the points that satisfy the inequality, meaning any point you choose from the shaded area would make the inequality true. In a sense, you've now mapped out the entire park where any location is a valid solution.