Solve the conjunction -2<or=a+3<8
Here is my result :5<or = a<5
Graph y>or=2x+1
Given the pair of inequalities
y>x+1
y<3/2x+3 graph their solution set
Thanks Danielle
For the first one, it's probably just a typo, but it should be -5<or=a<5
For the graph, the equation is in y=mx+b form, so b is the y-intercept. Since b=1, that means that the line hits the y-axis at (0,1). You need another point to make the line, so find the x-intercept by making y=0. Solve for x and you should get -1/2, so the point where it crosses the x-axis is (-1/2,0). This is a solid line because it is <or= not just <. So, shade above the line and make sure the line is solid.
For the next problem, draw both lines but don't shade yet. Both of these lines should be dotted, because it is not <or= or >or=. Shade above the first line but only if it is also below the second line.
For the first problem, you have the conjunction -2 < a + 3 < 8. To solve this, you want to isolate the variable "a" in the middle.
Step 1: Subtract 3 from all parts of the inequality: -2 - 3 < a + 3 - 3 < 8 - 3
Simplifying, you get -5 < a < 5.
So the correct solution is -5 < a < 5, not 5 < or = a < 5.
Moving on to the next problem, you have the inequality y > 2x + 1. To graph this, you need to plot the points on the coordinate plane based on the given equation.
Step 1: Identify the y-intercept and mark the point (0, b). In this case, b = 1, so the y-intercept is (0, 1).
Step 2: Find the x-intercept by substituting y with 0 in the equation and solving for x. 0 > 2x + 1 → -1 > 2x → x < -1/2. So the x-intercept is (-1/2, 0).
Step 3: Plot the two points and draw a line connecting them. Since the inequality is y > 2x + 1 (greater than), the line should be dotted to represent that the points on the line are not included in the solution set.
Step 4: Shade the region above the line to indicate that all points above the line satisfy the inequality.
For the final problem, you have a pair of inequalities: y > x + 1 and y < (3/2)x + 3. To graph their solution set, follow these steps:
Step 1: Begin by graphing each inequality separately. The first inequality, y > x + 1, represents a line with a slope of 1 and a y-intercept of (0, 1). Since the inequality uses a greater than symbol, the line should be dotted, and you should shade the region above the line.
Step 2: The second inequality, y < (3/2)x + 3, represents a line with a slope of 3/2 and a y-intercept of (0, 3). The line should also be dotted, but this time you should shade the region below the line.
Step 3: The solution set is the region where both inequalities overlap. Shade this region where the shaded regions from steps 1 and 2 overlap.
Remember to label your axes so that the graph is clear and complete.
I hope this explanation helps you better understand how to solve and graph these types of problems. Let me know if you have any further questions!