Which is the graph of the linear inequality [tex]6 x+2 y\ \textgreater \ -10[/tex]?
Answer (1)
Rewrite the inequality in slope-intercept form: -3x - 5"> y > − 3 x − 5 .
Graph the line y = − 3 x − 5 as a dashed line.
Test the point (0, 0) in the inequality: -5"> 0 > − 5 , which is true.
Shade the region above the dashed line. The final answer is the graph of the inequality -10"> 6 x + 2 y > − 10 .
Explanation
Understanding the Inequality We are given the linear inequality -10"> 6 x + 2 y > − 10 and we want to find its graph.
Converting to Slope-Intercept Form 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.
Isolating y To isolate y , we first subtract 6 x from both sides of the inequality: -6x - 10"> 2 y > − 6 x − 10
Solving for y Next, we divide both sides by 2: -3x - 5"> y > − 3 x − 5
Graphing the Line Now we have the inequality in slope-intercept form. The slope is − 3 and the y-intercept is − 5 . Since the inequality is strict (i.e., "> > and not ≥ ), we will graph the line y = − 3 x − 5 as a dashed line.
Determining the Shaded Region To determine which side of the line to shade, we can test a point that is not on the line. A common choice is the origin ( 0 , 0 ) . Plugging in x = 0 and y = 0 into the inequality -3x - 5"> y > − 3 x − 5 , we get: -3(0) - 5"> 0 > − 3 ( 0 ) − 5
-5"> 0 > − 5
This is true, so we shade the region above the dashed line.
Final Answer Therefore, the graph of the linear inequality -10"> 6 x + 2 y > − 10 is the region above the dashed line y = − 3 x − 5 .
Examples
Linear inequalities are used in various real-world applications, such as determining the feasible region in linear programming problems. For example, a company might use linear inequalities to determine the optimal production levels of two different products, given constraints on resources like labor and materials. By graphing the inequalities, the company can visualize the set of possible production levels that satisfy all the constraints.
