Solve the compound inequality. Graph the solution.
–2 ≤ 2x – 4 < 4
To solve the compound inequality -2 ≤ 2x - 4 < 4, we need to solve two separate inequalities.
First, let's solve the left inequality -2 ≤ 2x - 4:
-2 + 4 ≤ 2x - 4 + 4
2 ≤ 2x
2/2 ≤ 2x/2
1 ≤ x
Next, let's solve the right inequality 2x - 4 < 4:
2x - 4 + 4 < 4 + 4
2x < 8
2x/2 < 8/2
x < 4
Combining the solutions, we have 1 ≤ x < 4. This means that x must be greater than or equal to 1 but less than 4.
To graph the solution, we will draw a number line and shade the region between 1 and 4, including 1 but not including 4. The shaded region represents the solution to the compound inequality.
Here is the graph of the solution on a number line:
```
---[=======)--------
1 4
```
The shaded region starts at 1 and goes up to, but does not include, 4.
To solve the compound inequality –2 ≤ 2x – 4 < 4, we will first solve the two inequalities separately and then intersect their solutions.
1) Starting with the first inequality:
–2 ≤ 2x – 4
Add 4 to both sides to isolate 2x:
–2 + 4 ≤ 2x – 4 + 4
2 ≤ 2x
Divide both sides by 2:
2/2 ≤ 2x/2
1 ≤ x
So, the solution to the first inequality is x ≥ 1.
2) Moving on to the second inequality:
2x – 4 < 4
Add 4 to both sides to isolate 2x:
2x – 4 + 4 < 4 + 4
2x < 8
Divide both sides by 2:
2x/2 < 8/2
x < 4
Therefore, the solution to the second inequality is x < 4.
Now, we need to find the intersection of the two solutions.
The solution for the compound inequality is x ≥ 1 and x < 4.
To graph the solution on a number line, we start by marking the points x = 1 and x = 4 as open circles (since they are not included in the solution).
On the number line, we draw a solid line from x = 1 to x = 4, highlighting this range as the solution.
The graph represents all the values of x that satisfy the given compound inequality.