Graph 7x-9y< 5 -2x+4y> 5
The problem presented is a system of linear inequalities in two variables, x and y. The question requires graphing the inequality "7x - 9y < 5" and the inequality "-2x + 4y > 5" on a coordinate plane. The solution to this problem will be the region that satisfies both inequalities at the same time. The graphing involves finding the boundary lines by treating the inequalities as equalities (7x - 9y = 5 and -2x + 4y = 5), and then shading the appropriate side of these lines according to the direction of the inequalities. The intersection of the shaded regions will represent the set of points that satisfy both original inequalities.
$7 x - 9 y < 5$$- 2 x + 4 y > 5$
Answer
Solution:
Step 1: Graph the inequality $7x - 9y < 5$.
Step 1.1: Convert to slope-intercept form, $y = mx + b$.
Step 1.1.1: Isolate $y$.
Step 1.1.1.1: Subtract $7x$ from both sides: $-9y < -7x + 5$.
Step 1.1.1.2: Divide by $-9$, remembering to reverse the inequality: $y > \frac{7x}{9} - \frac{5}{9}$.
Step 1.2: Identify the slope $m$ and y-intercept $b$.
Step 1.2.1: Slope $m = \frac{7}{9}$ and y-intercept $b = -\frac{5}{9}$.
Step 1.3: Draw a dashed boundary line for $y = \frac{7}{9}x - \frac{5}{9}$ and shade above it.
Step 2: Graph the inequality $-2x + 4y > 5$.
Step 2.1: Convert to slope-intercept form, $y = mx + b$.
Step 2.1.1: Isolate $y$.
Step 2.1.1.1: Add $2x$ to both sides: $4y > 2x + 5$.
Step 2.1.1.2: Divide by $4$: $y > \frac{1}{2}x + \frac{5}{4}$.
Step 2.2: Identify the slope $m$ and y-intercept $b$.
Step 2.2.1: Slope $m = \frac{1}{2}$ and y-intercept $b = \frac{5}{4}$.
Step 2.3: Draw a dashed boundary line for $y = \frac{1}{2}x + \frac{5}{4}$ and shade above it.
Step 3: Superimpose both graphs on the same coordinate plane.
Step 4: The solution region is where the shaded areas of both inequalities overlap.
Knowledge Notes:
Inequalities and Graphs: When graphing inequalities, a dashed line is used for '<' or '>' (non-inclusive), and a solid line for '鈮? or '鈮? (inclusive). The graph is shaded above the line for '>' or '鈮? and below for '<' or '鈮?.
Slope-Intercept Form: The slope-intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. This form is useful for graphing because it easily shows the rate of change and the starting point on the y-axis.
Solving Inequalities: When solving inequalities, similar steps to solving equations are followed, but with one important exception: if you multiply or divide by a negative number, the direction of the inequality sign must be reversed.
Graphing Multiple Inequalities: When graphing two inequalities on the same coordinate plane, the solution set is the intersection of the shaded regions for each inequality.
Slope and Y-Intercept: The slope of a line indicates its steepness and direction (positive slope goes up, negative slope goes down). The y-intercept is the point where the line crosses the y-axis. These two values are pivotal in graphing linear functions and inequalities.
