Problem 41 Sketch the graph of the inequali... [FREE SOLUTION]

Chapter 6: Problem 41

Sketch the graph of the inequality. $$2 y>1$$

Short Answer

Expert verified

The graph of the inequality $2y > 1$ is a region shaded above the line y = 0.5 in the Cartesian coordinate system. This line is dashed to represent that points along the line do not satisfy the inequality, and the x-coordinate can take any real value.

Step by step solution

Isolate y

The first step is to isolate y. This can be done by dividing each side of the inequality $2y > 1$ by 2, giving $y > 0.5$.

Sketch the Graph

This inequality can now be graphed. Begin by drawing a horizontal line at y = 0.5 on a two-dimensional Cartesian coordinate system. As the inequality is strictly 'greater than', it implies that the line at y = 0.5 should be dotted or dashed to indicate that it is not included in the solution set. Then, because y is greater than 0.5, the area above this line should be shaded to indicate that all points in this region satisfy the inequality.

Interpret the Graph

Every point that lies in the shaded area above the line y = 0.5 represents a number pair (x, y) that satisfies the inequality. It should be noted that as there is no relation given between x and y in the original inequality, the x-coordinate can take any real value for a given point (x, y) where y > 0.5.

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Key Concepts

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

Cartesian Coordinate System

The Cartesian Coordinate System is a way of plotting points in a plane using pairs of numbers known as coordinates. With this system, you have two axes that intersect at a right angle: the horizontal axis (x-axis) and the vertical axis (y-axis). Every point on this plane has a precise location given by an ordered pair (x, y).

Here's how it works:

The x-axis represents the horizontal direction.
The y-axis represents the vertical direction.
The point where these axes intersect is called the origin (0, 0).

This system allows you to plot points and draw shapes by using the coordinates as a guide. It's widely used in graphing equations and inequalities. When graphing the inequality from our exercise, you'll use this system to locate the line y = 0.5 and shade in the appropriate region.

Inequality Notation

Inequality notation is a way to express a range of values that a variable can take, instead of just one value. For the inequality $2y > 1$, we first isolate y to get $y > 0.5$. This tells us that y can be any number greater than 0.5.

Here's a breakdown:

$>$ indicates 'greater than'.
$<$ indicates 'less than'.
$\geq$ indicates 'greater than or equal to'.
$\leq$ indicates 'less than or equal to'.

In our specific inequality, $y > 0.5$ means any value of y that is more than 0.5 makes the inequality true. In graphing, this affects whether lines are drawn solid (if the number itself is included, e.g., $\geq$) or dashed (if it is not included, e.g., $>$).

Solutions of Inequalities

When solving inequalities, we're looking for all possible values of a variable that make the inequality true. For the inequality $y > 0.5$, every solution is a point (x, y) where the y-value is greater than 0.5.

Here's how you interpret it on a graph:

Draw the boundary line. For $y > 0.5$, this line is $y = 0.5$
Use a dashed line, since $y = 0.5$ itself is not included in the solution.
Shade the region above the dashed line. This represents all the $y$ values greater than 0.5.

Every point within this shaded region is a solution to the inequality. Since nothing in the original problem restricts the values of x, any real value of x is possible. This means an infinite number of solutions exist, all satisfying the condition $y > 0.5$.

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91影视

Sketch the graph of the inequality. $$2 y>1$$

Short Answer

Step by step solution

Isolate y

Sketch the Graph

Interpret the Graph

Key Concepts

Cartesian Coordinate System

Inequality Notation

Solutions of Inequalities

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