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Chapter 7: Problem 12

Sketch the graph of the inequality. $$y>4 x-3$$

Short Answer

Expert verified
A graph including a dashed line representing the equation \(y = 4x - 3\) with the area above the line shaded. This shaded region represents the solutions to the inequality \(y > 4x - 3\).

Step by step solution

01

Plot the line \(y = 4x - 3\)

Firstly, arrange the inequality in the form of a linear equation: \(y = 4x - 3\) which has a slope of 4 and y-intercept of -3. Then, plot this line on a graph using dashed lines to indicate that the solutions to the inequality do not include the points on the line. The y-intercept will be (-3), and from there, use the slope to identify another point. The slope is 4, which means for every 1 unit increase in x, y increases by 4 units. So another point on the line is (1,1).
02

Shade the correct region

The inequality is \(y > 4x - 3\), meaning that the solutions to the inequality are all values of y that are greater than the values of y on the line \(y = 4x - 3\). This means we should shade the area above the line.
03

Re-examine the inequality and the graph

Look back at the original inequality and the sketch. Make sure the line and the shading match the inequality. Specifically, ensure that the line is indeed dashed and the shading is above the line.

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Key Concepts

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

Linear Equations
Linear equations are equations that create straight lines when plotted on a graph. They have the general form \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept. Linear equations can express a direct relationship between two variables, x and y. In the context of inequalities, linear equations form the boundary line that separates different regions of the graph.
  • The inequality \( y > 4x - 3 \) transforms into the linear equation \( y = 4x - 3 \) for the purpose of graphing.
  • This line divides the coordinate plane into two parts, with each side representing a different set of solutions.
Once the line is plotted, the inequality \( y > 4x - 3 \) tells us that the solution set includes all points above this line. This is why we use a dashed line to illustrate that points on the line are not included in the solution of the inequality.
Coordinate Plane
A coordinate plane is a two-dimensional surface where we can plot points, lines, and curves to represent mathematical concepts visually. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
  • The intersection of these axes is called the origin and has coordinates (0,0).
  • Any point on the coordinate plane is represented by an ordered pair (x, y), where x denotes the horizontal position and y denotes the vertical position.
In the exercise, the primary task is to sketch the graph of the equation \( y = 4x - 3 \) on the coordinate plane. This line serves as the boundary for the inequality \( y > 4x - 3 \). The entire process involves determining the y-intercept and using the slope to plot the line correctly on this system of reference.
Slope and Intercept
The slope and intercept of a line are crucial components in understanding and graphing linear equations. They tell us the direction and starting point of the line, respectively.
  • The slope, often denoted by \( m \), is a measure of the line's steepness. It tells us how much y increases (or decreases) as x increases by one unit.
  • In the equation \( y = 4x - 3 \), the slope is 4. This indicates that for every additional unit of x, y increases by 4 units.
  • The y-intercept is the value of y when x is 0. It is represented by \( b \) in the equation \( y = mx + b \).
  • For our equation, the y-intercept is -3, meaning the line crosses the y-axis at the point (0, -3).
By knowing the slope and intercept, you can easily identify another point on the line. Start at the y-intercept, and use the slope to find the next point 鈥� in this case, move 1 unit right and 4 units up to get to point (1,1).

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Most popular questions from this chapter

Write the partial fraction decomposition of the rational expression. Then assign a value to the constant \(a\) to check the result algebraically and graphically. $$\frac{1}{x(x+a)}$$

Sketch the region determined by the constraints. Then find the minimum and maximum values of the objective function (if possible) and where they occur, subject to the indicated constraints. Objective function: \(z=3 x+2 y\) Constraints: $$\begin{array}{r}x \geq 0 \\\y \geq 0 \\\5 x+2 y \leq 20 \\\5 x+y \geq 10\end{array}$$

After graphing the boundary of the inequality \(x+y<3,\) explain how you decide on which side of the boundary the solution set of the inequality lies.

Find values of \(x, y,\) and \(\lambda\) that satisfy the system. These systems arise in certain optimization problems in calculus, and \(\lambda\) is called a Lagrange multiplier. $$\left\\{\begin{aligned} 2 x-2 x \lambda &=0 \\ -2 y+\lambda &=0 \\ y-x^{2} &=0 \end{aligned}\right.$$

Find the equation of the circle $$x^{2}+y^{2}+D x+E y+F=0$$ that passes through the points. To verify your result, use a graphing utility to plot the points and graph the circle. $$(0,0),(5,5),(10,0)$$
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