Question

Ality 2x+3y>6, then tion to the given ineq

Answer 1

The given inequality is
`\(2x + 3y > 6\).` It represents a region in the coordinate plane where points
`\((x, y)\)` satisfy the inequality. To interpret this geometrically, the points lying above the corresponding line
`\(2x + 3y = 6\)` are solutions to the inequality.

Explanation:

The inequality
`\(2x + 3y > 6\)` is a linear inequality in two variables
`(\(x\) and \(y\))`. To understand the solution set graphically, we can first consider the corresponding equation
`\(2x + 3y = 6\).`

This equation represents a line in the coordinate plane. When the inequality is strict
`(\( > \)),` it means the solutions are not on the line but rather on one side of it.

To graphically represent the solution, rearrange the equation to
`\(3y > 6 - 2x\),` then divide by 3 to get
`\(y > 2 - (2)/(3)x\).` This expression represents the slope-intercept form of the line, where the coefficient of
`\(x\)` is the slope and the constant term is the y-intercept. The line
`\(2x + 3y = 6\)` is then graphed, and the region above the line
`(\(y > 2 - (2)/(3)x\)`) represents the solution set for the inequality.

In conclusion, the solution to the given inequality
`\(2x + 3y > 6\)` lies in the region above the line
`\(2x + 3y = 6\)` in the coordinate plane. Graphical representation helps in visualizing the solutions and understanding the geometric interpretation of the inequality.