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Chapter 5: Problem 7

Graph each inequality. $$y>2 x-1$$

Short Answer

Expert verified
To solve the inequality, graph the line \(y = 2x - 1\) with a dashed line. Then, test a point not on the line (e.g., the origin). If the point satisfies the inequality, shade that side. If not, shade the other side. In this case, the region above the line is shaded.

Step by step solution

01

Plot the Line

Start by ignoring the inequality and graph the line \(y = 2x - 1\). This is a straight line with a slope (gradient) of 2 and a y-intercept of -1. The y-intercept is where the line crosses the y-axis. To help in drawing the line, one can mark another point by stepping up 2 units (the slope) to the right 1 unit from the y-intercept.
02

Draw a Dashed Line

Because the inequality symbol is '>' and not '>=', the line should be drawn as dashed, not solid. This indicates that the points on the line are not part of the solution to the inequality.
03

Shade the Correct Side of the Line

To shade the correct side of the line, pick any point not on the line and test it in the inequality. If it satisfies the inequality, shade that side of the line. If not, shade the other side. For example, using the point (0,0) in the inequality, we see 0 > 2*0 - 1; 0 > -1 is true, so the area above the line (including (0,0)) should be shaded.

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Key Concepts

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

Linear Inequalities
Linear inequalities describe a relationship between two expressions where one side is not necessarily equal to the other, but either greater than or less than. For instance, inequality like \( y > 2x - 1 \) implies that for any given value of \( x \), \( y \) is always greater than the line defined by \( 2x - 1 \).
This helps us understand not just a single solution, but a range of solutions on a coordinate plane.
Unlike equalities, inequalities represent an entire region of possible solutions, rather than just a line or point.
  • They can dictate how a group of variables behave in relation to each other under specific conditions.
  • They are represented graphically by lines and shaded regions that indicate solution areas.
Slope-Intercept Form
When dealing with linear equations and inequalities, one common form you will encounter is the slope-intercept form. It is written as \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.
The slope \( m \) dictates the angle of the line and how steep it is. In our example with the inequality \( y > 2x - 1 \), the slope is 2, meaning the line rises 2 units for every 1 unit it moves to the right.
  • The y-intercept \( b \) shows where the line crosses the y-axis.
  • This is important as it provides a starting point for graphing the line, such as \( -1 \) in our case.
Understanding this form allows us to quickly interpret and translate a linear inequality into its visual representation on a graph.
Graph Interpretation
Reading and understanding a graph requires interpreting visual cues from the plotted line and shaded areas. With the line from \( y = 2x - 1 \) plotted, you can deduce much about the inequality.
  • The slope tells the direction and steepness while the y-intercept sets the starting point.
  • Knowing whether to use a dashed or solid line helps determine if points on the line are part of the solution. In our inequality, the dashed line means the equation \( y = 2x - 1 \) itself isn鈥檛 included in the solution.
The next part involves shading, which translates the inequality into a visual form and shows areas that satisfy the inequality.
Inequality Graph Shading
Shading a graph helps show the range of possible solutions for a linear inequality. For \( y > 2x - 1 \), shading helps us understand all the points \((x, y)\) where \( y \) is greater than the expression on the other side of the inequality.
To determine which side to shade:
  • Pick a test point not on the line, like \( (0,0) \).
  • Substitute it into the inequality. If true, shade the side of the line containing that point. In this example, since \(0 > -1\) is true, we shade above the line, where \( (0,0) \) is located.
Shading correct regions is crucial because it visually indicates where all potential solutions lie within our coordinate system."

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write the partial fraction decomposition of each rational expression. $$ \frac{10 x^{2}+2 x}{(x-1)^{2}\left(x^{2}+2\right)} $$

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