Question
The temperature at which a material will remain a liquid is represented by the inequality |2t – 3| ≤ 17. Which of the following graphs represents the solution set for this inequality?
Answers
GPT-4o mini
To solve the inequality \( |2t - 3| \leq 17 \), we start by breaking it down into two separate inequalities based on the definition of absolute value.
The inequality \( |x| \leq a \) can be rewritten as:
\[
-a \leq x \leq a.
\]
For our specific inequality, this means:
\[
-17 \leq 2t - 3 \leq 17.
\]
Next, we can deal with these two inequalities separately.
1. **From \(2t - 3 \geq -17\):**
\[
2t - 3 \geq -17 \\
2t \geq -17 + 3 \\
2t \geq -14 \\
t \geq -7.
\]
2. **From \(2t - 3 \leq 17\):**
\[
2t - 3 \leq 17 \\
2t \leq 17 + 3 \\
2t \leq 20 \\
t \leq 10.
\]
Combining these results, we have:
\[
-7 \leq t \leq 10.
\]
In interval notation, the solution set can be expressed as:
\[
[-7, 10].
\]
To graph this solution set, you would represent it on a number line with a solid line between -7 and 10, including the endpoints -7 and 10 which are represented by closed circles.
Look for a graph that illustrates this closed interval from -7 to 10 on a number line.
The inequality \( |x| \leq a \) can be rewritten as:
\[
-a \leq x \leq a.
\]
For our specific inequality, this means:
\[
-17 \leq 2t - 3 \leq 17.
\]
Next, we can deal with these two inequalities separately.
1. **From \(2t - 3 \geq -17\):**
\[
2t - 3 \geq -17 \\
2t \geq -17 + 3 \\
2t \geq -14 \\
t \geq -7.
\]
2. **From \(2t - 3 \leq 17\):**
\[
2t - 3 \leq 17 \\
2t \leq 17 + 3 \\
2t \leq 20 \\
t \leq 10.
\]
Combining these results, we have:
\[
-7 \leq t \leq 10.
\]
In interval notation, the solution set can be expressed as:
\[
[-7, 10].
\]
To graph this solution set, you would represent it on a number line with a solid line between -7 and 10, including the endpoints -7 and 10 which are represented by closed circles.
Look for a graph that illustrates this closed interval from -7 to 10 on a number line.