In Exercises 69–70, rewrite each inequality in the system without absolute value bars. Then graph the rewritten system in rectangular coordinates. |x|≤2, |y|≤3

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Step 1: Understand the meaning of absolute value inequalities. The inequality |x| ≤ 5 means that the distance of x from 0 on the number line is at most 5. This can be rewritten as a compound inequality: -5 ≤ x ≤ 5.

Step 2: Similarly, the inequality |y| ≤ 7 means that the distance of y from 0 on the number line is at most 7. This can be rewritten as -7 ≤ y ≤ 7.

Step 3: Combine the two inequalities to form a system without absolute values: -5 ≤ x ≤ 5 and -7 ≤ y ≤ 7.

Step 4: To graph this system in rectangular coordinates, draw the vertical boundary lines x = -5 and x = 5, and the horizontal boundary lines y = -7 and y = 7.

Step 5: The solution set is the rectangular region bounded by these lines, including the boundary lines themselves, since the inequalities are 'less than or equal to'. Shade this rectangle on the coordinate plane.

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주요 개념

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Absolute Value Inequalities

Absolute value inequalities express constraints on the distance of a variable from zero. For example, |x| ≤ 5 means x lies between -5 and 5. Rewriting these inequalities without absolute value bars involves splitting them into two inequalities, such as -5 ≤ x ≤ 5.

Graphing Inequalities in Rectangular Coordinates

Graphing inequalities in rectangular coordinates involves shading the region of the coordinate plane that satisfies the inequality. For inequalities like -5 ≤ x ≤ 5 and -7 ≤ y ≤ 7, the solution is a rectangle bounded by vertical and horizontal lines at these values.

Systems of Inequalities

A system of inequalities requires finding the common solution region that satisfies all inequalities simultaneously. For the given system, the solution is the intersection of the regions defined by each inequality, resulting in a rectangular area on the coordinate plane.