Problem 8 - Thinking Mathematically

Question

Graph each linear inequality. \(3 x+4 y \leq-12\)

Step-by-Step Solution

Answer

The graph should be a downwards sloping line, with a y-intercept at -3 and slope of -3/4. The line itself should be solid and everything below it should be shaded.

Step 1: Converting Inequality to Slope-Intercept Form

First, we'll start by converting \(3x + 4y \leq -12\) to slope-intercept form (y = mx+b), by isolating y. Begin by subtracting 3x from both sides to get: \(4y \leq -3x -12\), then divide both sides by 4, which results to: \(y \leq -\frac{3}{4}x -3\)

Step 2: Graphing the Line

Now we graph the corresponding line \(y = -\frac{3}{4}x -3\). First plot the y-intercept, which is -3. Then use the slope, -3/4, to find another point. Since the slope is -3/4, that means we can go down 3 units and to the right 4 units from the y-intercept to find another point. Plot this and connect to draw the line.

Step 3: Shading the Area

The inequality \(y \leq -\frac{3}{4}x -3\) includes an equal to sign, it means the line itself is part of the solution, so the line is solid. As for shading, since the inequality sign is “less than or equal to”, it means we must shade the area below the line

Step 4: Checking the Solution

Finally, you can check if the drawing is correct by taking a point not on the line, For example, zero point (0,0). If it satisfies the inequality, then the shading is correct. If not, the the shading will be on the other side.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common ways to write linear equations. It is represented by the equation y = mx + b, where m stands for the slope of the line and b indicates the y-intercept, which is the point where the line crosses the y-axis.

Understanding how to convert a linear inequality like 3x + 4y \(\leq\) -12 into slope-intercept form is crucial for graphing. As demonstrated in the exercise, we isolate y to find y \leq -\frac{3}{4}x -3, where -\frac{3}{4} represents the slope and -3 is the y-intercept. By identifying these components, one can graph the line effectively, providing a visual representation of all the points that satisfy the inequality.

Inequality Notation

Inequality notation expresses the relationship between two values, where they are not necessarily equal. The symbols used include \( < \), \( > \), \( \leq \), and \( \geq \). Each symbol has a specific meaning:

• \( < \) means 'less than',
• \( > \) is 'greater than',
• \( \leq \) stands for 'less than or equal to', and
• \( \geq \) indicates 'greater than or equal to'.

When graphing linear inequalities, the inequality sign tells us whether to use a solid or dashed line, and where to shade the graph. For instance, in our exercise, the \( \leq \) sign indicates that the line is included in the solution set (solid line) and that the area below the line should be shaded.

Shading Inequalities

Shading is an essential step in graphing inequalities. It visually represents the set of all possible solutions to the inequality. When the inequality is \( \leq \) or \( < \), you shade below the line because you are including all the y-values that are less than the y-value of the line at any given x-value. Conversely, if the inequality is \( \geq \) or \( > \), you shade above the line.

In our problem, the inequality \(y \leq -\frac{3}{4}x -3\) requires a shade below the line because the inequality indicates 'less than or equal to'. The inclusion of the equal sign (\leq) tells us to draw a solid line to include all points on the line as part of the solution.

Solving Inequalities

Solving inequalities involves finding all values that make the inequality true. Unlike equations, where we seek one or more specific solutions, with inequalities, we often end up with a range of solutions. Checking these solutions is an important step to verify the shaded region on the graph. We can pick a test point, usually (0,0) if it's not on the inequality line, and substitute it into the inequality.

If the test point satisfies the inequality, then we have shaded the correct region. For example, plugging (0,0) into our inequality \(y \leq -\frac{3}{4}x -3\) gives us 0 \leq -3 which is not true, indicating that the point (0,0) is not in the shaded region, and we shaded correctly below the line. This step is a great exercise for students to confirm their graphed solutions.