solve the compound inequalities show all math work arriving at the solution display a simple line graph 1-x>-3 or 5x-1>19
1-x > -3
1+3 > x
x < 4
5x-1 > 19
5x > 20
x > 4
Looks like all numbers except x=4.
To solve the compound inequalities 1-x>-3 or 5x-1>19, we will solve each inequality separately and then combine the solutions.
1) 1-x > -3:
First, let's isolate x by adding x and subtracting 1 from both sides of the inequality:
1 - x + x > -3 + x - 1
1 > -3 + x
Next, simplify the right side:
1 > x - 2
Now, isolate x by adding 2 to both sides:
1 + 2 > x - 2 + 2
3 > x
So, the solution for the first inequality is x < 3.
2) 5x - 1 > 19:
First, let's isolate x by adding 1 to both sides and then divide both sides by 5:
5x - 1 + 1 > 19 + 1
5x > 20
x > 20/5
x > 4
So, the solution for the second inequality is x > 4.
Now, we can combine the solutions:
The solution for the compound inequalities 1-x > -3 or 5x-1 > 19 is x < 3 or x > 4.
To display the solution on a simple line graph, we will mark the points x = 3 and x = 4 with open circles to indicate that they are not included in the solution. Then, we shade the area to the left of 3 and to the right of 4 on the number line to represent the solutions x < 3 or x > 4.
Here is a simple line graph to represent the solution:
```
-
|
|
----
|
--|----|----
| 3 4
```
The shaded areas represent the solution x < 3 or x > 4.
To solve the compound inequalities and graph the solution, we will address each inequality separately using the following steps:
First inequality: 1 - x > -3
Step 1: Subtract 1 from both sides of the inequality to isolate the variable.
1 - 1 - x > -3 - 1
-x > -4
Step 2: To eliminate the negative sign in front of the variable, multiply both sides by -1. However, since we are multiplying by a negative number, we need to flip the inequality.
-1 * -x < -4 * -1
x < 4
Second inequality: 5x - 1 > 19
Step 1: Add 1 to both sides of the inequality to isolate the variable.
5x - 1 + 1 > 19 + 1
5x > 20
Step 2: Divide both sides by 5 to find the value of x.
5x / 5 > 20 / 5
x > 4
Now we have our two solutions:
x < 4
x > 4
To graph the solution on a number line, we will use an open circle to represent the numbers that do not satisfy the inequality, and a shaded region to represent the numbers that do satisfy the inequality.
Let's draw a number line with 0 at the center and mark the value of 4 on it. Then, we will represent the solutions on the number line:
-------------o----- 4 ----------------
Since in the first inequality, x is less than 4 (x < 4), we will shade the left side of 4 as the solution:
<===========[ 4 )----------------------------
In the second inequality, x is greater than 4 (x > 4), so we will shade the right side of 4 as the solution:
<===========( 4 ]=======================>
This graph represents the solution to the compound inequality. The shaded region includes all real numbers greater than 4 or less than 4.