Problem 21 - Algebra 2

Question

Solve each system of inequalities by graphing. $$ \begin{array}{l}{x+y \leq 2} \\ {4 x^{2}-y^{2} \geq 4}\end{array} $$

Step-by-Step Solution

Answer

Shade below the line \( x + y = 2 \) and outside the hyperbola \( 4x^2 - y^2 \geq 4 \); solution is the overlap.

Step 1: Graph the First Inequality

For the inequality \( x + y \leq 2 \), we first consider the equation \( x + y = 2 \). To graph this, find two points on the line, such as when \( x = 0 \), then \( y = 2 \) (point (0, 2)), and when \( y = 0 \), then \( x = 2 \) (point (2, 0)). Draw a solid line through these points since the inequality is \( \leq \). Shade the area below this line, as it includes points where \( x + y < 2 \).

Step 2: Graph the Second Inequality

Consider the second inequality \( 4x^2 - y^2 \geq 4 \). First, restate it as an equation: \( 4x^2 - y^2 = 4 \), which represents a hyperbola. The equation can be rewritten as \( y^2 = 4x^2 - 4 \). To graph, find the intercepts: if \( x = 0 \), \( y^2 = -4 \), which is not possible, so no y-intercept; for \( y = 0 \), \( 4x^2 = 4 \), thus \( x = rac{1}{2} ext{ or } x = -rac{1}{2} \). The hyperbola opens along the x-axis. Use test points such as \( (1, 0) \) to check which side to shade; if it satisfies the inequality, \( (1, 0) ightarrow 4(1)^2 - (0)^2 \geq 4 \), it's part of the solution, so shade outside curves.

Step 3: Identify the Overlapping Region

The solution to the system of inequalities is the region where the shaded areas from both inequalities overlap. Check for the graphical intersection of the shaded region from Step 1 with the hyperbolic region from Step 2.

Step 4: Verify a Test Point

Select a point within the intersection region, such as \( (1,1) \), and verify both inequalities. For the first inequality, \( 1 + 1 = 2 \), which is true. For the second inequality, \( 4(1)^2 - (1)^2 = 3 ot\geq 4 \), so adjust and test surrounding points or ensure shading accuracy. Continue testing points until finding one fulfilling both.

Key Concepts

Graphing Inequalities

When tackling a system of inequalities, graphing each one helps visualize the solutions. Let's start by focusing on how to graph inequalities.
For the inequality \( x + y \leq 2 \), you first graph the boundary line given by the equation \( x + y = 2 \). To achieve this:

• Find two points on the line: when \( x = 0 \), \( y = 2 \) (point (0, 2)), and when \( y = 0 \), \( x = 2 \) (point (2, 0)).
• Draw a solid line through these points. The line is solid because the inequality is \( \leq \), including the line itself.
• Since the inequality is less than or equal to, shade the region below the line. This shaded area represents all possible solutions where \( x + y \leq 2 \).

Shading is crucial as it indicates valid solution regions. Picking test points on either side of the line can also determine which side to shade.

Inequality Solutions

Finding solutions to a system of inequalities involves determining where the inequalities overlap on the graph. In our system, we have two inequalities to solve:

• \( x + y \leq 2 \)
• \( 4x^2 - y^2 \geq 4 \)

After graphing each inequality, the solution to the system is where the shaded areas intersect. Start by:

• Graphing each inequality as per the procedures given previously.
• Checking the overlap region carefully. The valid solutions lie within this overlapping shaded region.
• Verifying the overlap using a test point within this region, such as \( (1, 1) \). Check if both inequalities are satisfied for this point to confirm that the intersection is correct.

Finding the overlap ensures you capture all combinations of \( x \) and \( y \) that solve both inequalities. This concept is central to comprehensive solution strategies.

Hyperbola Graphing

One of the equations, \( 4x^2 - y^2 = 4 \), describes a hyperbola. Graphing hyperbolas can be tricky, so understanding their nature helps:

• A hyperbola in this format opens along the x-axis, differentiating it from other conic sections like parabolas or ellipses.
• For this specific hyperbola, note that it has branches because \( \pm \sqrt{4x^2 - 4} = y \), affecting how the entire graph looks.
• Calculate intercepts to get a sense of orientation: for \( x = 0 \), \( y^2 = -4 \), indicating no y-intercept; for \( y = 0 \), \( x = \pm 1 \).
• Use test points like \( (1, 0) \) to determine shading. Since \( 4(1)^2 - 0^2 \geq 4 \), it confirms the point is within the solution space, describing the area outside the branches.

Graphing hyperbolas requires attention to detail and understanding of their geometric properties to ensure accurate representation and shading directions, integral in finding a solution.