The Correct Answer and Explanation is:
Based on the blurry text in the image, the problem is to solve the compound inequality: x + 8 ≥ 9 AND 2x – 1 < 5.
Correct Answer: 1 ≤ x < 3
Explanation
The problem presented is a compound inequality, which consists of two separate inequalities joined by the word “AND”. For a value of x to be a solution, it must satisfy both inequalities simultaneously. The process involves solving each inequality independently and then finding the intersection, or the overlapping portion, of their respective solution sets.
First, let’s solve the inequality x + 8 ≥ 9. The goal is to isolate the variable x on one side. To achieve this, we can subtract 8 from both sides of the inequality. This operation gives us x + 8 – 8 ≥ 9 – 8, which simplifies to x ≥ 1. This result means that any valid solution for x must be a number that is greater than or equal to 1. In interval notation, this solution set is represented as [1, ∞).
Next, we solve the second inequality, 2x – 1 < 5. This is a two step process. First, we need to isolate the term containing x, which is 2x. We do this by adding 1 to both sides of the inequality, resulting in 2x – 1 + 1 < 5 + 1, which simplifies to 2x < 6. The second step is to completely isolate x by dividing both sides by 2. This gives us x < 3. This result tells us that any valid solution for x must also be a number strictly less than 3. In interval notation, this solution set is written as (-∞, 3).
Finally, we must combine these two solutions. Since the inequalities are connected by “AND”, we are looking for the numbers that are both greater than or equal to 1 AND less than 3. Visualizing this on a number line, we find the intersection of the two sets. The final solution is all numbers between 1 and 3, including 1 but not including 3. This is written as the compound inequality 1 ≤ x < 3.
