Solve the inequality. Graph the solution set and write it in interval notation.
- 4 < 2(x - 2) <= 11
Write the answer in interval notation.
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(Type your answer in interval notation. Use integers or fractions for any numbers in the expression.)
To solve the inequality -4 < 2(x - 2) <= 11, we can simplify and solve each part separately.
Starting with 2(x - 2), we distribute the 2: 2(x - 2) = 2x - 4.
The inequality becomes:
-4 < 2x - 4 ≤ 11
Adding 4 to each part, we have:
0 < 2x ≤ 15
Dividing each part by 2, we get:
0 < x ≤ 7.5
So the solution set for the inequality is the interval (0, 7.5].
Graphically, this means that x is greater than 0 but less than or equal to 7.5, so the solution would be represented by a shaded line segment from 0 to 7.5, with 0 being an open circle and 7.5 being a closed circle.
The solution set in interval notation is: (0, 7.5]
Step 1: Distribute the 2 on both sides of the inequality:
-4 < 2x - 4 ≤ 11
Step 2: Add 4 to all parts of the inequality:
-4 + 4 < 2x - 4 + 4 ≤ 11 + 4
0 < 2x ≤ 15
Step 3: Divide all parts of the inequality by 2:
0/2 < 2x/2 ≤ 15/2
0 < x ≤ 7.5
Step 4: Graph the solution:
To graph the solution, we draw an open circle at 0 and a solid circle at 7.5 on a number line. Then we shade the line between the two points, indicating all the values of x that satisfy the inequality.
o •-------------------
-∞ 0 7.5 ∞
Step 5: Write the solution in interval notation:
The solution set, in interval notation, is (0, 7.5]
To solve the inequality -4 < 2(x - 2) ≤ 11, we need to follow these steps:
Step 1: Distribute the 2 on the right side of the inequality:
-4 < 2x - 4 ≤ 11
Step 2: Add 4 to all parts of the inequality:
-4 + 4 < 2x - 4 + 4 ≤ 11 + 4
0 < 2x ≤ 15
Step 3: Divide all parts of the inequality by 2:
0/2 < 2x/2 ≤ 15/2
0 < x ≤ 15/2
The solution to the inequality is 0 < x ≤ 15/2.
Now, let's graph the solution set and write it in interval notation:
To graph the solution set, we'll use a number line. Plot an open circle at 0 (since it's not included in the solution) and darken the line to the right up to the point 15/2 (including it since it satisfies the inequality).
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Now, to write the solution set in interval notation, we can represent it as (0, 15/2]. The opening parenthesis indicates that 0 is not included, and the closing bracket means that 15/2 is included.