Problem 41 - Intermediate Algebra

Question

Graph each inequality. $$ 0.3 x+0.4 y \geq-1.2 $$

Step-by-Step Solution

Answer

Graph the line through points (-4, 0) and (0, -3) and shade the region including (0, 0).

Step 1: Convert the Inequality to Equality

Start by converting the inequality into an equation for graphing the boundary line. Replace \( \geq \) with \( = \):\[ 0.3x + 0.4y = -1.2 \]

Step 2: Determine Intercepts

Find the x-intercept by setting \( y = 0 \) in the equation:\[ 0.3x + 0.4(0) = -1.2 \]\[ 0.3x = -1.2 \]\[ x = -4 \]For the y-intercept, set \( x = 0 \):\[ 0.3(0) + 0.4y = -1.2 \]\[ 0.4y = -1.2 \]\[ y = -3 \]

Step 3: Plot Intercepts and Draw the Line

Plot the points \((-4, 0)\) and \((0, -3)\) on a coordinate plane. Draw a solid line through these points (since the inequality is \( \geq \) which includes equality).

Step 4: Determine the Shaded Area

Choose a test point that is not on the line, such as \((0, 0)\), to determine which side of the line to shade. Substitute \( x = 0 \) and \( y = 0 \) into the original inequality:\[ 0.3(0) + 0.4(0) \geq -1.2 \]\[ 0 \geq -1.2 \]This statement is true, so shade the region that includes the point \((0, 0)\).

Key Concepts

Boundary Line

To start graphing an inequality such as \(0.3x + 0.4y \geq -1.2\), it's essential to first graph a boundary line. This line helps in dividing the plane into separate regions. You will replace the inequality symbol \( \geq \) with an equality symbol \( = \) to create the boundary line equation:

• Transform the inequality into an equation: \(0.3x + 0.4y = -1.2\).
• This line represents all the points \((x, y)\) where the equation holds exactly true.

In cases where the inequality includes \( \geq \) or \( \leq \), like in our example, the boundary line is drawn as a solid line. This indicates that points on the line satisfy the inequality. If the inequality were \( > \) or \( < \), a dashed line would be used, indicating that points on the line do not satisfy the inequality.

Shaded Region

The shaded region on a graph is crucial since it showcases all possible solutions to an inequality. This region is determined only after you've drawn the boundary line. For the inequality \(0.3x + 0.4y \geq -1.2\), the correct region to shade can be identified using a test point, which we will discuss later.Once you choose and test a point, such as \((0, 0)\), determine its validity by substituting it into the inequality:

• Substitute into the inequality: \(0.3(0) + 0.4(0) \geq -1.2\) results in \(0 \geq -1.2\).
• This is true, so the side of the boundary line that includes \((0, 0)\) is shaded.

The shaded area visually represents the set of all points that satisfy the inequality. Always verify with a test point to ensure accurate shading.

Intercepts

Intercepts are where the boundary line crosses the axes. Finding these points is essential for accurately drawing the line. Let's explore how to find the x- and y-intercepts for \(0.3x + 0.4y = -1.2\).

Finding the X-Intercept

• Set \(y = 0\) in the equation and solve for \(x\): \(0.3x + 0.4(0) = -1.2\).
• After simplifying: \(0.3x = -1.2\) gives \(x = -4\).
• The x-intercept is therefore \((-4, 0)\).

Finding the Y-Intercept

• Set \(x = 0\) in the equation and solve for \(y\): \(0.3(0) + 0.4y = -1.2\).
• After simplifying: \(0.4y = -1.2\) gives \(y = -3\).
• The y-intercept is therefore \((0, -3)\).

These intercepts \((-4, 0)\) and \((0, -3)\) provide specific points to plot on the graph to assist in drawing your boundary line.

Test Point Method

The test point method is a smart and straightforward way to identify the correct region to shade. For the inequality \(0.3x + 0.4y \geq -1.2\), after drawing the boundary line, choose a test point that is not on the boundary line, generally an easy point like \((0, 0)\).

• Choose \((0, 0)\) since it's simple to calculate and often distinct from the line.
• Substitute it into the original inequality: \(0.3(0) + 0.4(0) \geq -1.2\).
• Calculate to get \(0 \geq -1.2\), which is true.
• This confirms that the region containing \((0, 0)\) satisfies the inequality.

If the test point makes the inequality true, then shade the region containing this point. If it makes the inequality false, then shade the opposite side. This ensures the shaded area accurately represents all solutions to the inequality.