Problem 26 - Elementary Algebra

Question

Graph the linear equations and inequalities. $$ \frac{-5 y}{8} \leq 4 $$

Step-by-Step Solution

Answer

Now simplify the inequality: $$ y \geq -\frac{32}{5} $$ #tag_title#Step 2: Identify the slope (m) and y-intercept (b)#tag_content# In this case, our inequality is in the form of y ≥ mx + b, where m = 0 (since there is no x term) and b = -\frac{32}{5}. #tag_title#Step 3: Plot the y-intercept on the graph and use the slope to determine a second point#tag_content# Plot the y-intercept (-\frac{32}{5}) on the y-axis. Since the slope is 0, the line is horizontal and remains at a constant height of -\frac{32}{5} across the x-axis. #tag_title#Step 4: Draw the line (or dashed line), including the appropriate shading for the inequality#tag_content# Since the inequality is 'greater than or equal to' (≥), we will draw a solid horizontal line at y = -\frac{32}{5}. Next, we will shade the region above the line since y-values are greater in that area. The graph of the inequality $$y \geq -\frac{32}{5}$$ has a solid horizontal line at y = -\frac{32}{5} and shading above the line.

Step 1: Rewrite the inequality in slope-intercept form (y = mx + b)

Our inequality is given as: $$ \frac{-5y}{8} \leq 4 $$ First we need to isolate y on the left side of the inequality. To do this, multiply both sides by \(-\frac{8}{5}\). Remember to flip the inequality sign since we are multiplying by a negative number: $$ y \geq -\frac{8}{5}(4) $$

Step 2: Identify the slope (m) and y-intercept (b)#tag_content

In this case, our inequality is in the form of y ≥ mx + b, where m = 0 (since there is no x term) and b = -\frac{32}{5}. #tag_title

Step 3: Plot the y-intercept on the graph and use the slope to determine a second point#tag_content

Plot the y-intercept (-\frac{32}{5}) on the y-axis. Since the slope is 0, the line is horizontal and remains at a constant height of -\frac{32}{5} across the x-axis. #tag_title

Step 4: Draw the line (or dashed line), including the appropriate shading for the inequality#tag_content

Since the inequality is 'greater than or equal to' (≥), we will draw a solid horizontal line at y = -\frac{32}{5}. Next, we will shade the region above the line since y-values are greater in that area. The graph of the inequality $$y \geq -\frac{32}{5}$$ has a solid horizontal line at y = -\frac{32}{5} and shading above the line.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a special way of writing linear equations so they are easier to graph. It looks like this: \(y = mx + b\). Here, \(m\) is the slope and \(b\) is the y-intercept. This form shows us each line's steepness (slope) and where it crosses the y-axis. Recognizing these details can make graphing much simpler.

When dealing with inequalities, it's useful to rewrite them in a form similar to slope-intercept form. This helps visualize inequalities on a graph. For example, the inequality \(\frac{-5y}{8} \leq 4\) can be rewritten by isolating \(y\). This involves algebraic manipulation and ensures that the inequality's terms are arranged in a way that resembles the slope-intercept form. Remember, inequalities are very similar to linear equations, but they show a range of possible values instead of just one solution.

Graphing Inequalities

Graphing inequalities involves displaying a set of solutions on a coordinate plane instead of a solitary line or point. To do this effectively, especially with something like \(y \geq -\frac{32}{5}\), you can follow these steps:

• First, graph the equation 'as if it were an equality'. So plot \(y = -\frac{32}{5}\) on the y-axis as a horizontal line.
• Next, determine whether the inequality is 'greater than' or 'less than'; decide whether to shade above or below this line. In our example, \(y \geq -\frac{32}{5}\) means you would shade above the line.
• Decide if the line is solid or dashed:
  
  • Use a solid line when your inequality sign is either \(\leq\) or \(\geq\) to represent that points on the line satisfy the inequality.
  • Use a dashed line for \(<\) or \(>\), indicating that points on the line do not satisfy the inequality.

Graphing inequalities not only helps to visualize solutions but also reinforces understanding of which values satisfy the inequality.

Algebraic Manipulation

Algebraic manipulation involves adjusting an equation or inequality to make it easier to work with, often by isolating variables. It is a critical step when working with inequalities like \(\frac{-5y}{8} \leq 4\).

To isolate \(y\), as shown in our example, multiply both sides of the inequality by \(-\frac{8}{5}\). A crucial part of this step is remembering to flip the inequality sign anytime you multiply or divide by a negative number. Why do we flip the sign? Because multiplying or dividing by a negative reverses the direction of the inequality.

After manipulating the inequality, you reach a simplified form \(y \geq -\frac{32}{5}\). This version is much more manageable for graphing or further solving. Mastering algebraic manipulation is essential for solving all sorts of equations and inequalities, not just linear ones. It serves as a foundation for more advanced math topics.