Problem 33 - Algebra 2

Question

Graph each inequality on a coordinate plane. $$ 0.25 y-1.5 x \geq-4 $$

Step-by-Step Solution

Answer

The inequality \(0.25y-1.5x \geq -4\) is graphed by first converting to slope-intercept form to get \(y \geq 6x - 16\), then graphing a solid line through (0, -16) and (1, -10), and finally shading the region above the line.

Step 1: Convert to Slope-Intercept Form

Begin by isolating the y-variable on one side to convert the inequality into slope-intercept form, which is written as y = mx + b, where m is the slope and b is the y-intercept. Add \(1.5x\) to both sides: \(0.25y \- 1.5x + 1.5x \-4 + 1.5x \Rightarrow 0.25y \-4 + 1.5x\). Then divide every term by \(0.25\) to solve for y: \(y \- 4/0.25 + (1.5/0.25)x \Rightarrow y \-16 + 6x\).

Step 2: Graph the Boundary Line

The inequality has a 'greater than or equal to' sign, so the boundary line of the inequality is solid. Plot the y-intercept, which is at (0, -16). Then use the slope of 6 (rise over run) to plot another point. From the y-intercept, go up 6 units and right 1 unit to get the second point. Draw a solid line through these points; this is the boundary line.

Step 3: Shade the Appropriate Region

Since we have a 'greater than or equal to' inequality, we need to shade above the boundary line. This area represents all the points that satisfy the inequality \(0.25y-1.5x \geq -4\).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is a fundamental stepping stone in graphing inequalities. Typically expressed as y = mx + b, this form allows for the straightforward identification of the slope, m, and the y-intercept, b. In the slope-intercept form, m represents the rate of change, indicating how steep or flat the line is, and b specifies the point where the line crosses the y-axis.

In describing the exercise, we take the inequality 0.25y - 1.5x \[\geq\] -4 and manipulate it to resemble the slope-intercept layout. By adding 1.5x to both sides and dividing every term by 0.25, we achieve y \[\geq\] 6x - 16. Here the slope is 6, indicating a steep incline, and the y-intercept is -16, pinpointing the starting position on the graph.

Boundary Line

The boundary line acts as a defining edge for graphing inequalities. In our problem, the inequality includes a 'greater than or equal to' (\[\geq\]) symbol, prompting the need for a solid line. A solid line signifies that points on the line itself satisfy the inequality. This contrasts with a dashed line, which is used when the inequality does not include the equal part (Just \textgreater or \textless).

Here's how we graph it: Delineate the y-intercept (0, -16) on your coordinate plane. With the slope given as 6, identify another point by moving 6 units up and 1 unit right. Connect these points with a solid line to establish the boundary.

Shading Regions

Shading is a crucial step when graphing inequalities, as it illustrates which area of the graph contains solutions that satisfy the inequality. The rule of thumb is: If the inequality sign is 'greater than' (\textgreater) or 'greater than or equal to' (\[\geq\]), shade above the boundary line. Conversely, shade below for 'less than' (\textless) or 'less than or equal to' (\[\leq\]).

In our exercise, since the inequality is \[\geq\], we shade the region above the boundary line. This shaded space represents all the points where the value of y is greater than or equal to 6x - 16, which satisfies the inequality.

Solving Inequalities

Solving inequalities involves finding all the values of a variable that make the inequality true. The process is similar to solving equations, albeit the results often form a range of values instead of a singular solution. Important to remember is the 'inequality flipping rule': When you multiply or divide by a negative number, you must reverse the inequality sign.

In the given problem, solving the inequality meant converting it into slope-intercept form and subsequently graphing. However, solving could also entail plugging in values for x to find corresponding y values, affirming those that adhere to the inequality. Overall, solving inequalities sets the foundation for interpreting and representing real-world scenarios where conditions are not exact but rather fall within a certain spectrum.