y > −2x + 10
y > 1 over 2x − 2
There are various ways to graph inequalities, but one common method is to plot the corresponding equations of the boundary lines and then shade the region that satisfies the inequality. The boundary line of an inequality with a strict inequality sign (> or <) is a dashed line, while the boundary line of an inequality with a non-strict inequality sign (≥ or ≤) is a solid line.
To graph y > -2x + 10, we can start by finding two points on the line y = -2x + 10. One way is to set x = 0 and solve for y, and then set y = 0 and solve for x.
When x = 0: y = -2(0) + 10 = 10
When y = 0: 0 = -2x + 10 → x = 5
We can plot these points (0, 10) and (5, 0) and draw a dashed line through them.
To graph y > 1/2x - 2, we can again find two points on the line y = 1/2x -2.
When x = 0: y = 1/2(0) - 2 = -2
When y = 0: 0 = 1/2x - 2 → x = 4
We can plot these points (0, -2) and (4, 0) and draw a dashed line through them.
Now we need to shade the region that satisfies both inequalities. One way to do this is to pick a test point that is not on either boundary line and plug its coordinates into both inequalities.
For example, let's use (1, 1) as a test point. We plug it into the first inequality and get:
1 > -2(1) + 10 → 1 > 8
This is false, so we don't shade the region that includes (1, 1) for the first inequality.
Next, we plug (1, 1) into the second inequality and get:
1 > 1/2(1) - 2 → 1 > -1
This is true, so we shade the region that includes (1, 1) for the second inequality.
The shaded region for the second inequality is above the dashed line y = 1/2x - 2. Therefore, the solution to the system of inequalities is the region that satisfies both inequalities, which is the region above the line y = 1/2x - 2 but not above the line y = -2x + 10. We can express it as:
y > 1/2x - 2 and y ≤ -2x + 10
Solve the system of equations using substitution.
4x + 2y = 6
x = 2y + 4
We can solve the system of equations using substitution by solving one equation for one variable and substituting that expression into the other equation.
From the second equation, we can solve for x:
x = 2y + 4
Now we can substitute that expression for x into the first equation:
4x + 2y = 6
4(2y + 4) + 2y = 6
8y + 16 + 2y = 6
10y = -10
y = -1
Now we can substitute y = -1 back into either equation to solve for x. Let's use the second equation:
x = 2y + 4
x = 2(-1) + 4
x = 2
Therefore, the solution to the system of equations is (x, y) = (2, -1).
To find the solution to the system of inequalities, we need to solve each inequality separately and then find the region of overlap.
Let's start with the first inequality: y > -2x + 10.
Step 1: Graph the line y = -2x + 10.
To graph this line, we just need two points on the line. Let's find the x and y intercepts.
At x = 0, y = -2(0) + 10 = 10, so the point (0, 10) is on the line.
At y = 0, 0 = -2x + 10, which gives x = 5. So the point (5, 0) is also on the line.
Plot these two points and draw a straight line passing through them.
Step 2: Determine the inequality symbol (>) and the shading direction.
Since the inequality symbol is greater than (>) and not greater than or equal to (≥), the region above the line needs to be shaded.
Now let's move to the second inequality: y > 1/(2x - 2).
Step 3: Graph the line y = 1/(2x - 2).
To graph this line, we first need to determine where it is undefined. It is undefined when the denominator (2x - 2) equals 0.
So we set 2x - 2 = 0 and solve for x:
2x = 2
x = 1.
At x = 1, the line is undefined. But, since the inequality symbol is greater than (>) and not greater than or equal to (≥), we need to exclude this value from our solution.
Step 4: Determine the inequality symbol (>) and the shading direction.
Since the inequality symbol is greater than (>) and not greater than or equal to (≥), the region above the line needs to be shaded, excluding the value x = 1.
Now, let's find the region of overlap.
Step 5: Identify the overlapping region.
Since both inequalities have the shading direction above the lines, we need to find the overlapping region above both lines.
The overlapping region is the area above the line y = -2x + 10 and above the line y = 1/(2x - 2), excluding the value x = 1.
This can be represented graphically on a coordinate plane.
Please note that the solution depends on the context. If this is a mathematical problem, the solution would be represented graphically. If you are looking for specific values of x and y, further calculation or explanation may be needed.