In Exercises 39–45, graph each inequality. 3x - 4y > 12

Verified step by step guidance

1. Rewrite the inequality in slope-intercept form (y = mx + b) to make it easier to graph. Start by isolating the term involving y. Subtract 3x from both sides to get -4y > -3x + 12.
2. Divide every term in the inequality by -4 to solve for y. Remember, dividing or multiplying an inequality by a negative number reverses the inequality sign. This gives y < (3/4)x - 3.
3. Interpret the inequality y < (3/4)x - 3. The boundary line is y = (3/4)x - 3, which is a straight line with a slope of 3/4 and a y-intercept of -3. Since the inequality is '<', the region below the line will be shaded.
4. Graph the boundary line y = (3/4)x - 3. Use the slope (rise over run) and y-intercept to plot points. For example, start at (0, -3) (the y-intercept), then move up 3 units and right 4 units to plot another point. Connect these points with a dashed line because the inequality is strict ('<'), not inclusive ('≤').
5. Shade the region below the dashed line to represent the solution to the inequality. This shaded area includes all points (x, y) that satisfy y < (3/4)x - 3.

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They use symbols such as '>', '<', '≥', and '≤' to indicate whether one side is greater than, less than, or equal to the other. Understanding how to manipulate and interpret inequalities is essential for solving problems that involve ranges of values rather than fixed points.

Graphing Linear Inequalities

Graphing linear inequalities involves representing the solutions of the inequality on a coordinate plane. The boundary line is drawn based on the corresponding equation, and it is dashed if the inequality is strict ('>' or '<') and solid if it is inclusive ('≥' or '≤'). The area that satisfies the inequality is then shaded, indicating all the possible solutions.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as y = mx + b, where m represents the slope and b represents the y-intercept. This form is particularly useful for graphing because it allows for easy identification of the line's steepness and where it crosses the y-axis. Converting the inequality into this form can simplify the graphing process and help visualize the solution set.