91Ó°ÊÓ

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They use symbols such as <, >, ≤, and ≥. For example, in the inequality \( x + 3 < y \), it indicates that \( y \) is greater than \( x + 3 \). Unlike equations, inequalities do not show exact values, but rather a range of possible values. Inequalities are used in everyday situations like budgeting, measuring, and comparing data where exact values are not necessary or possible.

Graphing linear equations involves plotting points on a coordinate plane. A linear equation, such as \( y = x + 3 \), forms a straight line when graphed. The equation is in slope-intercept form: \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. To start graphing, you plot the y-intercept (the point where the line crosses the y-axis). Then, use the slope (rise over run) to plot additional points. Connect these points with a line. For inequalities like \( y > x + 3 \), use a dashed line to show that the values on the line are not included in the solution. Shade the region above the line to show where the inequality holds true.

The union of inequalities involves combining multiple inequalities to find a common solution set. In the given exercise, you need to find the solutions that satisfy either one of the inequalities or both. First, graph each inequality on the same coordinate plane. For instance, \( y > x + 3 \) and \( x > 3 \) are two inequalities. After graphing, shade the regions that represent the solutions for each inequality. The union of these inequalities is the area where either inequality holds true. This means any point within the union of the shaded regions satisfies at least one of the inequalities. In this example, the solution set is the area above the line \( y = x + 3 \) or to the right of the vertical line \( x = 3 \). By visualizing the union of inequalities, you can easily understand the range of possible solutions.