Problem 6 - Algebra 2

Question

Solve each system of inequalities by graphing. $$ \begin{array}{l}{x+y<4} \\ {9 x^{2}-4 y^{2} \geq 36}\end{array} $$

Step-by-Step Solution

Answer

The solution is the region where \( x + y < 4 \) overlaps with the outside of \( 9x^2 - 4y^2 \geq 36 \).

Step 1: Graph the Linear Inequality

The first inequality is \( x + y < 4 \). To graph it, we start by converting it to \( x + y = 4 \), a linear equation. Plot this line on the graph. Since the inequality is '<', use a dashed line to show that points on the line are not included. Identify that the region below the line (where \( x + y < 4 \) holds true) is the solution area for this inequality.

Step 2: Graph the Non-Linear Inequality

The second inequality is \( 9x^2 - 4y^2 \geq 36 \), which is an equation of a hyperbola \( 9x^2 - 4y^2 = 36 \). Dividing the entire inequality by 36 simplifies it to \( \frac{x^2}{4} - \frac{y^2}{9} \geq 1 \). Plot this hyperbola on the graph using a solid line to show that points on the curve are included. Select the region outside the branches of the hyperbola, because the inequality is '≥'.

Step 3: Identify the Solution Region

The solution to the system of inequalities is the area where the shaded regions from both inequalities overlap. This area displays all the points that simultaneously satisfy both conditions \( x + y < 4 \) and \( 9x^2 - 4y^2 \geq 36 \). Look for the intersection area of the graph from Step 1 and the graph from Step 2.

Key Concepts

Linear Inequalities

Linear inequalities are mathematical expressions involving a linear function, where the variables are compared using inequality symbols like `<, >, ≤,` or `≥`. Unlike linear equations, which provide a precise line graph, linear inequalities define a half-plane region.

• The general form of a linear inequality can be written as: \( ax + by < c \)
• Here, \( a \) and \( b \) are coefficients that determine the slope of the line, and \( c \) is the constant that shifts the line up or down the graph.

To solve a linear inequality:1. Begin by converting the inequality to an equation (e.g., replace `<` with `=`).2. Graph the equation as a straight line using the slope-intercept method or the x- and y-intercepts.3. Use a dashed line if the inequality symbol is `<` or `>`, meaning the points on the line are not included in the solution.4. Shade the region of the graph that satisfies the inequality. This region represents all possible solutions.

Graphing Inequalities

Graphing inequalities involves drawing the lines and regions that represent the possible solutions to an inequality or system of inequalities. This visual representation helps to identify the solution region for the given inequalities.

• Start by converting inequalities to equations for plotting.
• Draw the boundary line on the graph; use a dashed line for `<` or `>`, and a solid line for `≤` or `≥`.
• Identify which side of the line represents the solutions by testing a point not on the line.
• Shade the correct region to indicate all possible solutions of the inequality.

When graphing a system of inequalities, each inequality adds a constraint which limits the solution region further. The overlapping shaded areas of each inequality graph represent the set of solutions that satisfy the entire system.

Hyperbola

A hyperbola is a type of conic section formed by intersecting a plane with both nappes of a double cone. In terms of the exercise at hand, the inequality \( 9x^2 - 4y^2 \geq 36 \) represents a hyperbola.

• The standard form for a hyperbola is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) for horizontally oriented hyperbolas.
• In our case, reorganizing gives \( \frac{x^2}{4} - \frac{y^2}{9} \geq 1 \). The graph of the equation \( \frac{x^2}{4} - \frac{y^2}{9} = 1 \) is the boundary of the hyperbola.
• The solution region for the inequality \( 9x^2 - 4y^2 \geq 36 \) includes the space outside the hyperbola—this is indicated by the greater than or equal relation.

Hyperbolas have two disconnected curves or branches, and they extend indefinitely. Their asymptotes, seen as dashed lines, help to define the direction in which the branches spread.

Solution Region

The solution region of a system of inequalities is the intersection area where the solutions of all inequalities in the system overlap. This forms the set of points that satisfy each inequality simultaneously.

• For the inequality \( x + y < 4 \), the solution is the area below the line \( x + y = 4 \).
• For \( 9x^2 - 4y^2 \geq 36 \), the solution includes regions outside the hyperbola defined by \( 9x^2 - 4y^2 = 36 \).
• The solution region is found by graphing both conditions and locating where the shadings overlap.

It's crucial to correctly identify this overlapping shaded area on a graph because it represents all possible points (x, y) that satisfy both inequalities. This region is the complete solution to the system of inequalities given in the problem.