Problem 42 - Thinking Mathematically

Question

Write the given sentences as a system of inequalities in two variables. Then graph the system. The sum of the \(x\)-variable and the \(y\)-variable is at most 3 . The \(y\)-variable added to the product of 4 and the \(x\)-variable does not exceed 6 .

Step-by-Step Solution

Answer

The system of inequalities derived from the given sentences is \(x + y \leq 3\) and \(y + 4x \leq 6\). These inequalities form a triangular region in the first quadrant as the solution when graphed on the Cartesian plane.

Step 1: Formulating the Inequalities

From the first given sentence, we have: \(x + y \leq 3\). This implies that the sum of the \(x\)-variable and the \(y\)-variable is at most 3. From the second sentence, we infer: \(y + 4x \leq 6\). This states that the \(y\)-variable added to the product of 4 and the \(x\)-variable does not exceed 6.

Step 2: Graphing the Inequalities

To graph inequalities, we treat them as equations first. We draw the line for \(x + y = 3\) which will be a downward sloping line passing through points (3,0) and (0,3). As the inequality is \(x + y \leq 3\), the region we're interested in lies on or below this line. For \(y + 4x = 6\), we draw the line which will be a downward sloping line passing through points (1.5,0) and (0,6). As the inequality is \(y + 4x \leq 6\), the region of interest lies below this line. The solution to the system of inequalities is the intersection of the areas corresponding to each inequality, which happens to be the triangular area bounded by the x-axis, y-axis, and the line \(x + y = 3\).

Key Concepts

Graphing Inequalities

Graphing inequalities involves converting each inequality into a boundary line on the coordinate plane. This line helps divide the plane into regions that either satisfy or do not satisfy the inequality. Start by transforming the inequality into an equation to find the points that define the line. A simple rule is to replace the inequality sign with an equal sign to find these crucial points. For instance, with the inequality \(x + y \leq 3\), begin by plotting the line \(x + y = 3\). Identify points on this line, like (3,0) and (0,3), to draw it accurately.
Now it's important to find which side of the line represents the solution to the inequality. For a \(\leq\) or \(\geq\) sign, shade the area below or above the line respectively, as these symbols include the line itself in the solutions. Testing points like (0,0) can confirm which region satisfies the system. If plugging such a point into the inequality makes it true, shade towards that point. Finally, the overlapping shaded regions indicate where the solutions to all inequalities in the system lie.

Formulating Inequalities

When translating word problems into mathematical inequalities, start by identifying the variables involved. These variables represent unknown quantities that the problem is dealing with. In our case, the variables include \(x\) and \(y\). Next, interpret the problem context to form equations, focusing on key phrases related to maximum, minimum, at most, or at least statements.
For the given example, the sentence "The sum of the \(x\)-variable and the \(y\)-variable is at most 3" becomes \(x + y \leq 3\). "At most" translates to "less than or equal to." Similarly, "The \(y\)-variable added to the product of 4 and the \(x\)-variable does not exceed 6" turns into \(y + 4x \leq 6\). Formulation skills are crucial in keeping the inequalities faithful to the given problem statement, ensuring accuracy in further steps.

Algebra

Algebra is a vital tool for solving and visualizing inequalities as it provides a structured method for dealing with variables and equations. Solving algebraic inequalities involves finding the set of all possible values for the variables that make the inequality true. This often includes rearranging the inequality to better understand its structure or to isolate specific variables.
It's important to remember rules when manipulating inequalities. For instance, when multiplying or dividing by a negative number, the inequality sign must be flipped. Keeping these properties in mind helps to find the correct solutions. In practice, algebra involves not only the arithmetic of solving equations but also the visualization of solutions on graphs, where analysis of overlapping solutions in systems of inequalities occurs. Mastery of algebra helps interpret these intersections accurately, ensuring comprehensive solutions to the problems presented.