Graph the following inequality: [tex]x+y \leq 7[/tex]
Answer (1)
Rewrite the inequality: y ≤ − x + 7 .
Graph the line: y = − x + 7 (solid line).
Test the point (0, 0): 0 ≤ 7 (True).
Shade the region: below the line.
Explanation
Understanding the Inequality We are asked to graph the inequality x + y ≤ 7 . This means we need to find all the points ( x , y ) on the coordinate plane that satisfy this inequality.
Rewriting the Inequality First, let's rewrite the inequality in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. To do this, we isolate y on one side of the inequality:
x + y ≤ 7
y ≤ − x + 7
Graphing the Line Now, let's consider the equation y = − x + 7 . This is a linear equation representing a straight line with a slope of − 1 and a y-intercept of 7 . We can graph this line on the coordinate plane. Since the original inequality is y ≤ − x + 7 , we will draw a solid line to indicate that the points on the line are included in the solution.
Determining the Shaded Region Next, we need to determine which region of the coordinate plane to shade. To do this, we can choose a test point that is not on the line. A simple test point is ( 0 , 0 ) . Substitute this point into the original inequality:
0 + 0 ≤ 7
0 ≤ 7
Since 0 ≤ 7 is true, the region containing the point ( 0 , 0 ) satisfies the inequality. Therefore, we shade the region below the line.
Final Graph In summary, we graph the line y = − x + 7 as a solid line and shade the region below the line to represent all points ( x , y ) that satisfy the inequality x + y ≤ 7 .
Examples
Understanding linear inequalities is crucial in various real-world scenarios. For instance, consider a budget constraint where you have a limited amount of money to spend on two goods, like books and coffee. If 'x' represents the number of books and 'y' represents the number of cups of coffee, and each book costs $10 and each cup of coffee costs $2, the inequality 10 x + 2 y ≤ B (where B is your budget) represents the possible combinations of books and coffee you can afford. Graphing this inequality helps visualize all feasible spending options within your budget.
