Graph the inequality.
$y \geq-3 x+4$
Answer (1)
Graph the boundary line y = − 3 x + 4 using the y-intercept (0, 4) and x-intercept ( 3 4 , 0 ) .
Test the point (0, 0) in the inequality y g e − 3 x + 4 , which results in 0 g e 4 , which is false.
Shade the region above the line since (0, 0) does not satisfy the inequality.
Draw a solid line to indicate that the boundary line is included in the solution: y g e − 3 x + 4 .
Explanation
Understanding the Inequality We are asked to graph the inequality y g e − 3 x + 4 . This is a linear inequality, which means its graph will be a region in the coordinate plane bounded by a straight line.
Graphing the Boundary Line First, we need to graph the boundary line y = − 3 x + 4 . This is a line with a slope of -3 and a y-intercept of 4. To graph it, we can plot two points and draw a line through them. Let's find the y-intercept by setting x = 0 : y = − 3 ( 0 ) + 4 = 4 So, the y-intercept is (0, 4). Now, let's find the x-intercept by setting y = 0 :
0 = − 3 x + 4 3 x = 4 x = 3 4 So, the x-intercept is ( 3 4 , 0 ) .
Determining the Shaded Region Since the inequality is y g e − 3 x + 4 , we need to determine which side of the line to shade. We can choose a test point not on the line, such as (0, 0). Substitute this point into the inequality: 0 g e − 3 ( 0 ) + 4 0 g e 4 This is false, so the point (0, 0) does not satisfy the inequality. Therefore, we shade the region above the line.
Drawing the Graph Since the inequality is y g e − 3 x + 4 , the boundary line is solid, indicating that the points on the line are included in the solution.
Examples
Linear inequalities are used in various real-world scenarios, such as determining budget constraints. For example, if you have a budget of $50 and want to buy apples and bananas, where apples cost $2 each and bananas cost $1 each, the inequality 2 x + y l e 50 represents the possible combinations of apples (x) and bananas (y) you can buy within your budget. Graphing this inequality helps visualize all the feasible solutions.
