QuestionJuly 15, 2025

Air temperature decreases as altitude increases. If the ground temperature is 76^circ F then the air temperature x miles high is T=76-19x (a) Determine the altitudes x where the temperature T is between 2^circ F and 32^circ F. (b) Use an absolute value inequality to describe these altitudes. (a) Determine the altitudes x where the air temperature T is between 2^circ F and 32^circ F, inclusive. square leqslant xleqslant square (Type an integer or a simplified fraction.)

Air temperature decreases as altitude increases. If the ground temperature is 76^circ F then the air temperature x miles high is T=76-19x (a) Determine the altitudes x where the temperature T is between 2^circ F and 32^circ F. (b) Use an absolute value inequality to describe these altitudes. (a) Determine the altitudes x where the air temperature T is between 2^circ F and 32^circ F, inclusive. square leqslant xleqslant square (Type an integer or a simplified fraction.)
Air temperature decreases as altitude increases. If the ground temperature is 76^circ F then the air temperature x miles high is T=76-19x (a) Determine the altitudes x where the temperature T is between 2^circ F and 32^circ F. (b) Use an absolute value inequality to describe these altitudes. (a) Determine the altitudes x where the air temperature T is between 2^circ F and 32^circ F, inclusive. square leqslant xleqslant square (Type an integer or a simplified fraction.)

Solution
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Answer

\frac{44}{19} \leq x \leq \frac{74}{19} Explanation 1. Set up inequalities for temperature range Given T = 76 - 19x, set up the inequalities 2 \leq T \leq 32. 2. Solve inequality for lower bound For T \geq 2: \[ 76 - 19x \geq 2 \] \[ 76 - 2 \geq 19x \] \[ 74 \geq 19x \] \[ x \leq \frac{74}{19} \] 3. Solve inequality for upper bound For T \leq 32: \[ 76 - 19x \leq 32 \] \[ 76 - 32 \leq 19x \] \[ 44 \leq 19x \] \[ x \geq \frac{44}{19} \] 4. Combine results Combine the results from Steps 2 and 3 to find the interval: \[ \frac{44}{19} \leq x \leq \frac{74}{19} \] 5. Use absolute value inequality Express the interval using an absolute value inequality: \[ |x - \frac{59}{19}| \leq \frac{15}{19} \]

Explanation

1. Set up inequalities for temperature range<br /> Given $T = 76 - 19x$, set up the inequalities $2 \leq T \leq 32$.<br /><br />2. Solve inequality for lower bound<br /> For $T \geq 2$: <br />\[ 76 - 19x \geq 2 \]<br />\[ 76 - 2 \geq 19x \]<br />\[ 74 \geq 19x \]<br />\[ x \leq \frac{74}{19} \]<br /><br />3. Solve inequality for upper bound<br /> For $T \leq 32$: <br />\[ 76 - 19x \leq 32 \]<br />\[ 76 - 32 \leq 19x \]<br />\[ 44 \leq 19x \]<br />\[ x \geq \frac{44}{19} \]<br /><br />4. Combine results<br /> Combine the results from Steps 2 and 3 to find the interval:<br />\[ \frac{44}{19} \leq x \leq \frac{74}{19} \]<br /><br />5. Use absolute value inequality<br /> Express the interval using an absolute value inequality:<br />\[ |x - \frac{59}{19}| \leq \frac{15}{19} \]
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