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Chapter 8: Problem 13

In Exercises \(5-20,\) graph each linear inequality. $$3 x+4 y < 2$$

Short Answer

Expert verified
Graph the boundary line, use a dashed line, and shade the region below the line where \(3x+4y < 2\).

Step by step solution

01

Express as Equation

First, we convert the inequality into an equation to find the boundary line. For the inequality \(3x + 4y < 2\), the related equation is \(3x + 4y = 2\).
02

Find Intercepts

To graph the line, find the x-intercept and y-intercept of the equation. Set \(y = 0\) to find the x-intercept: \(3x + 4(0) = 2\) gives \(x = \frac{2}{3}\). Set \(x = 0\) to find the y-intercept: \(3(0) + 4y = 2\) gives \(y = \frac{1}{2}\).
03

Draw the Boundary Line

Plot the intercepts \((\frac{2}{3}, 0)\) and \((0, \frac{1}{2})\) on the graph and draw a dashed line through these points. The line is dashed because the inequality is strictly '<' and does not include the line itself.
04

Test a Point

Select a test point not on the line to determine which side of the line to shade. A common choice is \((0,0)\). Substitute into the inequality: \(3(0) + 4(0) < 2\), which simplifies to \(0 < 2\), a true statement. Therefore, shade the region containing the origin, which is below the line.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Boundary Line
When graphing a linear inequality, the "boundary line" is a crucial concept. The boundary line represents the corresponding equation, which helps visualize where the inequality stands in relation to the coordinate plane. For the inequality \(3x + 4y < 2\), we first transform it into its related equation: \(3x + 4y = 2\). This equation represents the boundary line.
  • It divides the coordinate plane into two regions.
  • Once plotted, this line visually separates the true and false solutions to the inequality.
In cases where the inequality does not include equality, such as '<' or '>', use a dashed line for the boundary. This indicates that the points on the line are not part of the solution.
x-intercept
The x-intercept is the point where the line crosses the x-axis. For the boundary line \(3x + 4y = 2\), we find the x-intercept by setting \(y = 0\) and solving for \(x\):\[3x + 4(0) = 2\]This simplifies to \(x = \frac{2}{3}\). So, the x-intercept is \(\left( \frac{2}{3}, 0 \right)\).
  • It provides a point that helps us plot the boundary line.
  • Understanding intercepts is key for accurately drawing linear graphs.
Intercepts are unbreakable pieces of graphing linear equations, ensuring that the graph aligns perfectly with the mathematical equation or inequality.
y-intercept
Similar to the x-intercept, the y-intercept is where the line crosses the y-axis. For the line \(3x + 4y = 2\), find the y-intercept by setting \(x = 0\) and solving for \(y\):\[3(0) + 4y = 2\]Solving this equation gives \(y = \frac{1}{2}\), so the y-intercept is \(\left( 0, \frac{1}{2} \right)\).
  • The y-intercept, along with the x-intercept, allows us to sketch the boundary line accurately.
  • Mastering intercept calculations bolsters overall understanding of plotting linear graphs.
Both intercepts create a strong foundation to accurately depict the behavior of the line associated with a linear inequality or equation.
Shading Region
Shading the region is a vital step in graphing inequalities. After plotting the boundary line, we must determine which side of the line represents the solution set. With the inequality \(3x + 4y < 2\), we need to decide which area to shade.
  • Select a test point not on the line, typically \((0,0)\), for ease of calculation.
  • Substitute this point into the original inequality. For \(3(0) + 4(0) < 2\), the statement \(0 < 2\) holds true.
As this point is true for the inequality, we shade the region that includes \((0,0)\), which lies below the dashed line. Remember, shading indicates all points that satisfy the linear inequality, highlighting the area of solutions within the coordinate plane.

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Most popular questions from this chapter

Apply a graphing utility to perform the indicated matrix operations. $$A=\left[\begin{array}{rrrr}1 & 7 & 9 & 2 \\\\-3 & -6 & 15 & 11 \\\0 & 3 & 2 & 5 \\\9 & 8 & -4 & 1\end{array}\right]$$ Find \(A^{-1}\)

Determine whether each of the following statements is true or false: A system of equations represented by a nonsquare coefficient matrix cannot have a unique solution.

You are asked to model a set of three points with a quadratic function \(y=a x^{2}+b x+c\) and determine the quadratic function. a. Set up a system of equations, use a graphing utility or graphing calculator to solve the system by entering the coefficients of the augmented matrix. b. Use the graphing calculator commands \([\text { STAT }]\) QuadReg to model the data using a quadratic function. Round your answers to two decimal places. $$(-9,20),(2,-18),(11,16)$$

In Exercises \(79-84,\) determine whether each statement is true or false. A dashed curve is used for strict inequalities.

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