Question

1. which inequality is represented by the graph below?

graph of a line and shaded region

1. $y \leq 2x - 3$
2. $y \geq 2x - 3$
3. $y \leq -3x + 2$
4. $y \geq -3x + 2$
5. which ordered pair is not in the solution set of $y > 2x + 1$?
6. $(1,4)$
7. $(1,6)$
8. $(3,8)$
9. $(2,5)$

explain or show how you know:
third problem (partially visible): a system of inequalities is graphed... which point is a solution to this system? (1,1), (2,-2), (1,8), (4,2)

1. first consider the system of equations $y = -\frac{1}{2}x + 1$ and $y = x - 5$. then consider the system of inequalities $y > -\frac{1}{2}x + 1$ and $y < x - 5$. when comparing the number of solutions in each of these systems, which statement is true?
2. both systems have an infinite number of solutions.
3. the system of equations has more solutions.
4. the system of inequalities has more solutions.
5. both systems have only one solution.

Explanation:

Question 1

Step1: Find the slope and y - intercept of the line

The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. From the graph, we can see that the line passes through points. Let's calculate the slope. If we take two points on the line, say, when \(x = 0\), we can find the y - intercept. The line in the graph has a slope \(m = 2\) (since for a rise of 2, the run is 1) and a y - intercept \(b=- 3\) (the line crosses the y - axis at \(y=-3\)). So the equation of the line is \(y = 2x-3\).

Step2: Determine the inequality (dashed or solid line and shading)

The line in the graph is solid (so the inequality is either \(\geq\) or \(\leq\)) and the shading is above the line. For a linear inequality \(y\geq mx + b\), the region above the line is shaded. So the inequality represented by the graph is \(y\geq2x - 3\).

Step1: Substitute the ordered pairs into the inequality \(y>2x + 1\)

• For the ordered pair \((1,4)\): Substitute \(x = 1\) and \(y = 4\) into \(y>2x + 1\). We get \(4>2(1)+1\), which is \(4>3\). This is true.
• For the ordered pair \((1,6)\): Substitute \(x = 1\) and \(y = 6\) into \(y>2x + 1\). We get \(6>2(1)+1\), which is \(6>3\). This is true.
• For the ordered pair \((3,8)\): Substitute \(x = 3\) and \(y = 8\) into \(y>2x + 1\). We get \(8>2(3)+1\), which is \(8>7\). This is true.
• For the ordered pair \((2,5)\): Substitute \(x = 2\) and \(y = 5\) into \(y>2x + 1\). We get \(5>2(2)+1\), which is \(5>5\). This is false.

Step1: Analyze the system of equations \(y=-\frac{1}{2}x + 1\) and \(y=x - 5\)

A system of two linear equations (a linear system) has either one solution (if the lines intersect), no solution (if the lines are parallel), or infinitely many solutions (if the lines are coincident). The two lines \(y =-\frac{1}{2}x+1\) and \(y=x - 5\) have different slopes (\(-\frac{1}{2}
eq1\)), so they intersect at exactly one point. So the system of equations has exactly one solution.

Step2: Analyze the system of inequalities \(y>-\frac{1}{2}x + 1\) and \(y

The inequality \(y>-\frac{1}{2}x + 1\) represents the region above the line \(y =-\frac{1}{2}x + 1\) and \(y

Step3: Compare the number of solutions

The system of equations has 1 solution and the system of inequalities has an infinite number of solutions. So the system of inequalities has more solutions.

Answer:

1. \(y\geq2x - 3\)

Question 2