Solve the given compound inequality. Present your answer in interval notation.
[tex]11 \leq 4 y+3<23[/tex]
The solution set is: $\square$
Answer (2)
Subtract 3 from all parts of the inequality: 11 − 3 ≤ 4 y + 3 − 3 < 23 − 3 , which simplifies to 8 ≤ 4 y < 20 .
Divide all parts of the inequality by 4: 4 8 ≤ 4 4 y < 4 20 , which simplifies to 2 ≤ y < 5 .
Express the solution in interval notation. Since y is greater than or equal to 2 and less than 5, the interval notation is [ 2 , 5 ) .
The solution set is [ 2 , 5 ) .
Explanation
Understanding the Problem We are given the compound inequality 11 ≤ 4 y + 3 < 23 . Our goal is to isolate y to find the solution set.
Isolating the Term with y First, we subtract 3 from all parts of the inequality to isolate the term with y : 11 − 3 ≤ 4 y + 3 − 3 < 23 − 3
This simplifies to: 8 ≤ 4 y < 20
Solving for y Next, we divide all parts of the inequality by 4 to solve for y : 4 8 ≤ 4 4 y < 4 20
This simplifies to: 2 ≤ y < 5
Expressing the Solution in Interval Notation The solution set includes all values of y that are greater than or equal to 2 and less than 5. In interval notation, this is represented as [ 2 , 5 ) . The square bracket indicates that 2 is included in the solution set, while the parenthesis indicates that 5 is not included.
Final Answer Therefore, the solution set for the compound inequality 11 ≤ 4 y + 3 < 23 is [ 2 , 5 ) .
Examples
Compound inequalities are useful in various real-life situations. For example, suppose you want to maintain a certain body temperature. Your body needs to be within a specific range to function correctly. Let's say your optimal body temperature should be between 36.5°C and 37.5°C. If your temperature goes outside this range, you might experience discomfort or health issues. Compound inequalities help define and maintain these safe and optimal ranges, ensuring systems or conditions stay within acceptable limits.
To solve the compound inequality 11 ≤ 4 y + 3 < 23 , we isolate y and simplify it to find 2 ≤ y < 5 . In interval notation, the solution set is [ 2 , 5 ) .
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