Which correctly describes how the graph of the inequality −4y − x ≥ 7 is shaded?
Graphing inequalities on a number line
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Learning how to find the maximum value of an objective
Wouldn’t it be great if we could only manufacture and sell an infinite number of units of a commodity and make an infinite amount of money? We know in business (and in everyday life) that we can’t just do anything because it seems reasonable that it will (unreasonably) achieve our target. Rather, we want to maximize or minimize a quantity within a set of constraints.
Consider this scenario: you’re flying from Chandler, Arizona, to San Diego, California. Your aim is to arrive in as little time as possible, so you’ll try to cut down on travel time. Around the same time, you’ll encounter more or less traffic on some sections of the journey, you’ll need to stop for gas at least once (unless you’re driving a hybrid), and you’ll almost certainly need to stop for a bathroom break if you have children. Though we’ve only listed a handful, all of these inequalities are represented by the symbols, >, [latex]le [/latex], and [latex]ge [/latex]. We can only concentrate on [latex]le [/latex] and [latex]ge [/latex] due to difficulties with strict inequalities (and >).
How to graph y = 3x + 2
The Cartesian plane (or rectangular coordinate system) allows you to map points to ordered pairs and ordered pairs to points. A one-to-one mapping from points in the plane to ordered pairs is known as a one-to-one mapping. The polar coordinate system offers a different way of converting points into ordered pairs. In this part, we’ll see how polar coordinates can be more useful than rectangular coordinates in certain situations.
Consider Figure 7.27 to find the coordinates of a point in the polar coordinate system. The coordinates of the point PP are Cartesian (x,y). (x, y) The length of the line segment connecting the origin to the point PP is r.r. It calculates the distance between the origin and PP. The angle measured between the positive xx-axis and the line segment is. This finding implies that the coordinate pair (x,y)(x,y) and the values rr and. have a natural correspondence. The polar coordinate system is built on this correspondence. Any point in the Cartesian plane has two values associated with it (hence the term ordered pair). Each point in the polar coordinate system has two values associated with it: rr and.
Smaller area enclosed by the circle x^2 +y^2 = 4 and the lines
We’ve learned how to solve one-variable inequalities. We’ll now examine inequalities in two variables. Two-variable inequalities have a wide range of applications. If you were running a company, for example, you would want your sales to exceed your costs in order to make a profit.
Can you recall that a one-variable inequality has several solutions? Any number greater than three is the solution to the inequality. On the number line, we demonstrated this by shading in the number line to the right of 3 and placing an open parenthesis at 3. Look into it (Figure).