Question

Solve and graph the inequalities below. There are several options for submitting your graphs. Please choose one of the options below:

A.)Create the graphs in a drawing program or using the whiteboard feature and attach them.
B.)Draw the graphs by hand and fax them or mail them to your instructor
C.)Describe your graphs using words as shown in the lesson.

1.) 2x > -6 and x – 4 < 3
2.) x + 5 > 2x + 1 and -4x < -8
3.) -3x < -6 or x + 5 < -2
4.) x - 2 > 2x + 1 or -2x - 2 < -10
please do work by hand a submit pictures

Answer 1

Inequalities are solved by isolating x and then graphed on a number line, with the solution sets determined by the direction of the inequality and any overlap between graphs. The 'or' condition allows for any number that satisfies either inequality, while 'and' requires numbers to satisfy both.

Step-by-step explanation:

To solve and graph the given inequalities, let's handle each one step by step.

1.) 2x > -6 and x – 4 < 3:
For the first inequality, divide both sides by 2 to isolate x, yielding x > -3. To graph this, you would draw a number line, place a circle on -3, and shade to the right to indicate all numbers greater than -3.
For the second inequality, add 4 to both sides to get x < 7. Graphically, you'll place a circle on 7 on the number line and shade to the left to represent all numbers less than 7. The solution set is where the two shaded areas overlap, between -3 and 7.

2.) x + 5 > 2x + 1 and -4x < -8:
Subtract x from both sides of the first inequality to find that 5 > x + 1, then subtract 1 from both sides to get 4 > x or x < 4. On the number line, place a circle at 4 and shade left. For the second inequality, divide both sides by -4, remembering to reverse the inequality sign, giving x > 2. Graph this with a circle at 2 and shade to the right. The two shaded areas do not overlap, indicating there is no solution set that satisfies both inequalities.

3.) -3x < -6 or x + 5 < -2:
Divide the first inequality by -3 to get x > 2 (after reversing the inequality due to dividing by a negative), and graph this on a number line with a circle at 2 shaded to the right. For the second inequality, subtract 5 from both sides to get x < -7, and graph with a circle at -7 shaded to the left. The solution set here includes all numbers less than -7 or greater than 2 because of the 'or' condition.

4.) x - 2 > 2x + 1 or -2x - 2 < -10:
For the first inequality, subtract x from both sides and then add 2 to obtain -1 > x or x < -1. Then, graph this with a circle at -1 and shade left. For the second inequality, add 2x to both sides and then add 2 to yield x > -4. Graph this with a circle at -4 and shade to the right. The solution set includes all numbers less than -1 and all numbers greater than -4 due to the 'or' condition, even though these ranges overlap.

Answer 2

Step-by-step explanation:

1. We have the inequalities 2x > -6 and x-4<3

So, we get,

2x > -6 ⇒ x > -3 and x-4<3 ⇒ x < 7.

Thus, the solution is given by x > -3 and x < 7 i.e. -3 < x < 7 given by figure 1.

2. We have the inequalities x + 5 > 2x + 1 and -4x < -8

So, it gives us,

x + 5 > 2x + 1 ⇒ 4 > x i.e. x < 4 and -4x < -8 ⇒ x > 2.

Then, the solution is given by x < 4 and x > 2 i.e. 2 < x < 4 given by figure 2.

3. We have the inequalities -3x < -6 or x+5 < -2

The solution is,

-3x < -6 ⇒ x > 2 or x+5 < -2 ⇒ x < -7.

Thus, the solution is given by x > 2 or x < -7 given by figure 3.

4. We have the inequalities x-2 > 2x+1 or -2x-2 < -10

We get,

x-2 > 2x+1 ⇒ x < -3 or -2x-2 < -10 ⇒ -2x < -8 ⇒ x > 4

So, the solution is given by x < -3 or x > 4 given by figure 4.