Problem 36 - Algebra A Combined Function

Question

Graph each inequality in two variables. $$ 3 x-y \leq 4 $$

Step-by-Step Solution

Answer

Shade above the solid line of \(y = 3x - 4\).

Step 1: Write the Inequality in Slope-Intercept Form

Start with the inequality \(3x - y \leq 4\). To write it in slope-intercept form \(y = mx + b \), solve for \(y\). Add \(y\) to both sides to get \(3x \leq y + 4\). Subtract \(4\) from both sides: \(3x - 4 \leq y\). Finally, rewrite it as \(y \geq 3x - 4\).

Step 2: Identify the Line for the Boundary

The boundary line is \(y = 3x - 4\), which you'll graph using a dashed or solid line. In this case, use a solid line because the inequality is \(\leq\), indicating that points on the line are included in the solution.

Step 3: Graph the Boundary Line

To graph the line, identify the y-intercept, which is -4. Start by plotting the y-intercept (0, -4) on the graph. Then use the slope, which is 3 (or \(\frac{3}{1}\)), to find another point: from (0, -4), go up 3 units and right 1 unit to arrive at (1, -1). Connect these points with a solid line.

Step 4: Shade the Solution Region

Since the inequality is \(y \geq 3x - 4\), shade the region above the line. This represents all points \((x, y)\) that make the inequality true.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a line is a popular way to express the equation of a line, especially useful for graphing. The formula is expressed as \( y = mx + b \). Here, \( m \) represents the slope of the line, telling you how steep the line is. The \( b \) in the formula is the y-intercept, the point where the line crosses the y-axis.
Writing the inequality \(3x - y \leq 4\) in slope-intercept form helps simplify graphing. First, we solve for \( y \). Add \( y \) to both sides:

• \(3x \leq y + 4\)

Subtract \( 4 \) from both sides, rearranging terms to isolate \( y \):

• \( y \geq 3x - 4 \)

Now, the inequality is in slope-intercept form (note the \( \geq \) sign meaning 'greater than or equal to'), making it clear that the line has a slope of 3 and intersects the y-axis at -4.

Boundary Line

In inequalities, the boundary line is straight from the slope-intercept form of the inequality. It acts as a delimiter for the solution region on the graph. For our inequality \( y \geq 3x - 4 \), the boundary line is \( y = 3x - 4 \), represented as a line on the graph.

• If the inequality sign is \( \leq \) or \( \geq \), the line is solid, indicating points on the line are part of the solution set.
• If the inequality were \( > \) or \( < \), you'd use a dashed line to show points on the line aren't included.

Our boundary line here is solid due to the \( \leq \) in the original inequality. This means all points on the line make the inequality true.

Solution Region

The solution region refers to the part of the graph that contains all possible solutions (\( x, y \) pairs) that satisfy the inequality. Once you graph the boundary line, you need to determine which side of the line contains the solutions.
Since our inequality after rearranging is \( y \geq 3x - 4 \), the solution region is above the boundary line. This means any point in the shaded area will satisfy the inequality. Remember:

• Shade above the line when the inequality is \( \geq \).
• Shade below the line for \( \leq \).

Shading helps visually represent all the potential solutions that align with the inequality rules.

Inequalities in Two Variables

An inequality in two variables, such as \(3x - y \leq 4\), can be graphically represented to better visualize the solutions. It involves finding a range of \( x \) and \( y \) values that make the inequality true, unlike equations which provide precise solutions.
The process begins by rearranging the inequality to the slope-intercept form. This helps plot the boundary line. The inequality symbol dictates whether the boundary line is solid or dashed and where the solution region is. Finally, shading the correct side of the boundary line illustrates all valid \((x, y)\) points.

• Use slope-intercept form \( y = mx + b \) to handle inequalities in two variables easily.
• Graphing makes complex solutions manageable and visually comprehensible.

Understanding these key steps helps solve and graphically interpret inequalities, providing a comprehensive solution view.