Systems of Equations and Inequalities In previous chapters we solved equations with one unknown or variable. We will now study methods of solving systems of equations consisting of two equations and two variables. Represent the Cartesian coordinate system and identify the origin and axes.
The Real Number Line. The most intuitive way is to use the real number line. If we draw a line, designate a point on the line to be zero, and choose a scale, then every point on the line corresponds uniquely to a real number, and vice versa: The real number line "respects" the order of the real numbers.
A bigger number will always be found to the right of a smaller number. We visualize a set on the real number line by marking its members. It is standard to agree on the following conventions: To include an endpoint, we "bubble it in. Here is the set of all real numbers greater than -2 and less than or equal to 5: The number -2 is excluded from the set, so you see an "empty bubble"; the number 5 is included in the set, so the bubble at 5 is "filled in.
The set does not need to be "connected.
The following is a description of the set of all real numbers with the exception of -1 and 2: Interval notation translates the information from the real number line into symbols. Our example becomes the interval -2,5].
To indicate that an endpoint is included, we use a square bracket; to exclude an endpoint, we use parentheses. Our example is written in interval notation as.
The infinity symbols " " are used to indicate that the set is unbounded in the positive or negative direction of the real number line. Therefore we always exclude them as endpoints by using parentheses. If the set consists of several disconnected pieces, we use the symbol for union " ": How could we write down in interval notation?
There are three pieces to consider: An interval such aswhere both endpoints are excluded is called an open interval. An interval is called closed, if it contains its endpoints, such as.
An unbounded interval such as is considered to be open; an interval such as is called closed even though it does not contain its right endpoint. The whole real line is considered to be both open and closed.Write an inequality for the graph.
Solve the inequality. Then graph your solution. 5. Write an inequality that describes this situation. Let c represent the number of cans of food that must be collected by the end of the third week for your class to meet or surpass your goal. b. Write the inequality that best describes each graph: 1) Inequality: 2) Inequality: 3) Inequality: 4) Write the inequality that best describes each graph: 1) Inequality: 2) Inequality: 3) Inequality: 4) Inequality: 5) Writing Inequalities.
For two-variable linear inequalities, the "equals" part is the graph of the straight line; in this case, that means the "equals" part is the line y = 2x + 3: Advertisement Now we're at the point where your book probably gets complicated, with talk of "test points" and such.
Then graph the inequality. 4 Write a verbal sentence that describes the. inequality. Then graph the inequality lt y lt 0; 7? t? 8; 4?
n lt 11; 5. Compound Inequalities in Real Life ; MOUNTAIN PLANT LIFE ; Write a compound inequality that describes the approximate elevation range for the type of plant life on Mount Rainier, a mountain peak in.
Solve each inequality. Graph the solution set on a number line. $(5 $(5 or $(5 or Write an inequality to represent the her femur bone.
b. If the length of a female skeleton ¶s femur is 50 centimeters, write and solve an absolute value inequality that describes the woman ¶s height in centimeters. $(5 a. b. Take the extra half a second, and write the solution correctly. The pattern for "greater than" absolute-value inequalities always holds: the solution is always in two parts.
Given the inequality | x | > a, the solution always starts by splitting the inequality into two pieces: x a. For instance: Solve | 2x – 3 | > 5.