Question 1

Write the statement as an absolute value inequality.
-The temperatures in a freeze-dry process in degrees Celsius roughly satisfy the inequality $| C + 99 |$ $\leq 64$. What range of temperatures corresponds to this inequality?
A) $\left[ - 35 ^ { \circ } \mathrm { C } , 163 ^ { \circ } \mathrm { C } \right]$
B) $\left[ - 163 ^ { \circ } \mathrm { C } , - 35 ^ { \circ } \mathrm { C } \right]$
C) $\left[ - 35 ^ { \circ } \mathrm { C } , 35 ^ { \circ } \mathrm { C } \right]$
D) $\left[ - 163 ^ { \circ } \mathrm { C } , 163 ^ { \circ } \mathrm { C } \right]$

Accepted Answer

The absolute value inequality $| C + 99 | \leq 64$ translates to two inequalities: $C + 99 \leq 64$ and $-C - 99 \leq 64$. Solving these gives $C \leq -35$ and $C \geq -163$, respectively, which corresponds to the range $[-163, -35]$ degrees Celsius.

Question 2

Solve the inequality. Write the solution set in interval notation.
-$$| x - 1 | \leq 0$$
A) $\{ - 1 \}$
B) $\{ 1 \}$
C) $( - \infty , 1 )$
D) $\varnothing$

Accepted Answer

The absolute value of an expression equals 0 if and only if the expression itself is 0. Therefore, $| x - 1 | \leq 0$ implies $x - 1 = 0$, which further simplifies to $x = 1$. This means the solution set consists of the single point $x = 1$, represented in interval notation as $\{ 1 \}$.

Question 3

Solve the inequality. Write the solution set in interval notation.
-$$\left| \frac { 4 x + 1 } { x - 7 } \right| \geq 0$$
A) $( - 7,7 )$
B) $( - \infty , 7 ) \cup ( 7 , \infty )$
C) $\left( - \infty , - \frac { 1 } { 4 } \right) \cup ( 7 , \infty )$
D) $\left( - \frac { 1 } { 4 } , 7 \right)$

Accepted Answer

The absolute value of any real number or algebraic expression is always greater than or equal to zero. The inequality $\left| \frac { 4 x + 1 } { x - 7 } \right| \geq 0$ is true for all $x$ except where the denominator is zero, which occurs at $x = 7$. Therefore, the solution set includes all real numbers except $x = 7$, represented by $( - \infty , 7 ) \cup ( 7 , \infty )$.

Question 4

Write the statement as an absolute value inequality.
-A real estate development consists of home sites that range in width from 53 to 105 feet and in depth from 125 to 187 feet. Using $ x $ as the variable in both cases, write absolute value inequalities that correspond to these ranges.
A) $ |x-79| \leq 26,|x-156| \leq 31 $
B) $ |B-52| \leq 53,|x-62| \leq 125 $
C) $ |B-53| \leq 52,|x-125| \leq 62 $
D) $ |x-26| \leq 79,|x-31| \leq 156 $

Accepted Answer

The midpoint of the width range is $\frac{53 + 105}{2} = 79$ and the difference from the midpoint to either end is $105 - 79 = 26$, so $|x - 79| \leq 26$. For depth, the midpoint is $\frac{125 + 187}{2} = 156$ and the difference to either end is $187 - 156 = 31$, so $|x - 156| \leq 31$.

Question 5

Write the statement as an absolute value inequality.
-In a milling operation, the thickness of the wood panels that can be produced satisfies the inequality $| \mathrm { x } - 1.88 | \leq 1.20$. What range of thicknesses corresponds to this inequality?
A) $[ 0.68,3.08 ]$
B) $[ 0.68,6.16 ]$
C) $[ 0.34,3.08 ]$
D) $[ 1.20,1.88 ]$

Accepted Answer

The absolute value inequality $|x - 1.88| \leq 1.20$ represents the range of values x can take such that their distance from 1.88 is at most 1.20 units. Solving the inequality, we get $0.68 \leq x \leq 3.08$, which corresponds to option A.

Question 6

Write the statement as an absolute value inequality.
-The high temperature on December 12 in Biloxi, MS ranges from $ 31^{\circ} \mathrm{F} $ to $ 89^{\circ} \mathrm{F} $. Using $ \mathrm{F} $ as the variable, write an absolute value inequality that corresponds to this range.
A) $ |F-29| \leq 60 $
B) $ |F-60| \leq 29 $
C) $ |F-31| \leq 58 $
D) $ |F-58| \leq 31 $

Accepted Answer

The midpoint of the range is 60, and the distance from the midpoint to either endpoint is 29, so the inequality is $ |F-60| \leq 29 $.

Question 7

Write the statement as an absolute value inequality.
-$\mathrm { m }$ is no more than 9 units from 8
A) $| m - 8 | \leq 9$
B) $| m - 8 | < 9$
C) $| m - 9 | < 8$
D) $| m - 8 | > 9$

Accepted Answer

The statement "m is no more than 9 units from 8" translates to the absolute value inequality $| m - 8 | \leq 9$, indicating that the distance between m and 8 is less than or equal to 9 units.

Question 8

Solve the inequality. Write the solution set in interval notation.
-$$| 2 x - 9 | \leq - 2$$
A) $\left( - \infty , \frac { 11 } { 2 } \right) \cup \left( \frac { 7 } { 2 } , \infty \right)$
B) $\left( \frac { 11 } { 2 } , \frac { 7 } { 2 } \right)$
C) $\left( - \infty , - \frac { 5 } { 2 } \right) \cup \left( - \frac { 9 } { 2 } , \infty \right)$
D) $\varnothing$

Accepted Answer

The absolute value of any real number is always non-negative, so the inequality $| 2x - 9 | \leq -2$ has no solution because the absolute value cannot be less than or equal to a negative number.

Question 9

Write the statement as an absolute value inequality.
-The number of non-text books read by college students ranges from 8 to 46 . Using $B$ as the variable, write an absolute value inequality that corresponds to this range.
A) $\mid$ B $- 27 \mid \leq 19$
B) $| \mathrm { B } - 8 | \leq 38$
C) $| B - 19 | \leq 27$
D) $| \mathrm { B } - 38 | \leq 8$

Accepted Answer

The midpoint of the range from 8 to 46 is $\frac{8 + 46}{2} = 27$, and the distance from this midpoint to either endpoint is $46 - 27 = 19$, so the correct absolute value inequality is $|B - 27| \leq 19$.

Question 10

Solve the inequality. Write the solution set in interval notation.
-$$| x + 5 | < 0$$
A) $\{ 5 \}$
B) $\{ - 5 \}$
C) $\varnothing$
D) $( - \infty , - 5 )$

Accepted Answer

The absolute value of any real number is always non-negative, meaning it is either positive or zero. Therefore, the inequality $| x + 5 | < 0$ has no solution, as the absolute value cannot be less than zero.

Question 11

Write the statement as an absolute value inequality.
-The average annual growth rate of Palmetto palms in inches satisfies the inequality $| \mathrm { x } - 4.72 | \leq 3.59$. What range of growth corresponds to this inequality?
A) $[ 1.13,8.31 ]$
B) $[ 1.13,16.62 ]$
C) $[ 0.57,8.31 ]$
D) $[ 3.59,4.72 ]$

Accepted Answer

The absolute value inequality $|x - 4.72| \leq 3.59$ describes the distance from $x$ to $4.72$ being less than or equal to $3.59$. Solving the inequality gives the range $1.13 \leq x \leq 8.31$, which corresponds to choice A.

Question 12

Write the statement as an absolute value inequality.
-$p$ is 2 units from 6
A) $| p - 6 | = 2$
B) $| p - 6 | < 2$
C) $| p | = 4$
D) $| p - 2 | = 6$

Accepted Answer

The statement "p is 2 units from 6" directly translates to the absolute value equation $| p - 6 | = 2$, indicating the distance between p and 6 is exactly 2 units.

Question 13

Solve the inequality. Write the solution set in interval notation.
-$$\left| x ^ { 4 } + 6 x ^ { 2 } + 9 \right| > 0$$
A) $\varnothing$
B) $( - 3,3 )$
C) $( - \infty , - 3 ) \cup ( 3 , \infty )$
D) $( - \infty , \infty )$

Accepted Answer

The expression inside the absolute value, $x^4 + 6x^2 + 9$, is a sum of squares and is always positive for all real x, except it equals 0 when $x^2 = -3$, which has no real solutions. Therefore, the absolute value of this expression is always greater than 0 for all real numbers, making the solution set all real numbers, or $( - \infty , \infty )$.

Question 14

Write the statement as an absolute value inequality.
-$\mathrm { x }$ is within 3 units of 2
A) $| x - 3 | < 2$
B) $| x - 2 | < 3$
C) $| 3 - x | < 2$
D) $| x - 2 | \leq 3$

Accepted Answer

The answer of Write the statement as an absolute value...

Question 15

Write the statement as an absolute value inequality.
-$\mathrm { z }$ is no less than 7 units from 12
A) $| z - 12 | \leq 7$
B) $| z - 7 | > 12$
C) $| z - 12 | \geq 7$
D) $| z - 12 | > 7$

Accepted Answer

The statement "z is no less than 7 units from 12" means the distance between z and 12 is at least 7, which is represented by $| z - 12 | \geq 7$.

Question 16

Solve the inequality. Write the solution set in interval notation.
-$$\left| x ^ { 4 } + 6 x ^ { 2 } + 9 \right| < 0$$
A) $( - \infty , \infty )$
B) $( - 3,3 )$
C) $( - \infty , - 3 ) \cup ( 3 , \infty )$
D) $\varnothing$

Accepted Answer

The expression inside the absolute value, $x^4 + 6x^2 + 9$, can be factored as $(x^2 + 3)^2$, which is always non-negative for all real numbers $x$. Since the square of a real number (including zero) is always non-negative, the expression inside the absolute value can never be negative. Therefore, the absolute value of a non-negative number cannot be less than 0. The solution set is empty, as there are no values of $x$ that satisfy the given inequality.

Quiz 2: Equations and Inequalities

Write the statement as an absolute value inequality. -The temperatures in a freeze-dry process in degrees Celsius roughly satisfy the inequality $| C + 99 |$ $\leq 64$ . What range of temperatures corresponds to this inequality?

Solve the inequality. Write the solution set in interval notation. - $| x - 1 | \leq 0$

Solve the inequality. Write the solution set in interval notation. - $\left| \frac { 4 x + 1 } { x - 7 } \right| \geq 0$

Write the statement as an absolute value inequality. -A real estate development consists of home sites that range in width from 53 to 105 feet and in depth from 125 to 187 feet. Using $x$ as the variable in both cases, write absolute value inequalities that correspond to these ranges.

Write the statement as an absolute value inequality. -In a milling operation, the thickness of the wood panels that can be produced satisfies the inequality $| \mathrm { x } - 1.88 | \leq 1.20$ . What range of thicknesses corresponds to this inequality?

Write the statement as an absolute value inequality. -The high temperature on December 12 in Biloxi, MS ranges from $31^{\circ} \mathrm{F}$ to $89^{\circ} \mathrm{F}$ . Using $\mathrm{F}$ as the variable, write an absolute value inequality that corresponds to this range.

Write the statement as an absolute value inequality. - $\mathrm { m }$ is no more than 9 units from 8

Solve the inequality. Write the solution set in interval notation. - $| 2 x - 9 | \leq - 2$

Write the statement as an absolute value inequality. -The number of non-text books read by college students ranges from 8 to 46 . Using $B$ as the variable, write an absolute value inequality that corresponds to this range.

Solve the inequality. Write the solution set in interval notation. - $| x + 5 | < 0$

Write the statement as an absolute value inequality. -The average annual growth rate of Palmetto palms in inches satisfies the inequality $| \mathrm { x } - 4.72 | \leq 3.59$ . What range of growth corresponds to this inequality?

Write the statement as an absolute value inequality. - $p$ is 2 units from 6

Solve the inequality. Write the solution set in interval notation. - $\left| x ^ { 4 } + 6 x ^ { 2 } + 9 \right| > 0$

Write the statement as an absolute value inequality. - $\mathrm { x }$ is within 3 units of 2

Write the statement as an absolute value inequality. - $\mathrm { z }$ is no less than 7 units from 12

Solve the inequality. Write the solution set in interval notation. - $\left| x ^ { 4 } + 6 x ^ { 2 } + 9 \right| < 0$

Quiz 2: Equations and Inequalities

Write the statement as an absolute value inequality. -The temperatures in a freeze-dry process in degrees Celsius roughly satisfy the inequality ∣C+99∣| C + 99 |∣C+99∣ ≤64\leq 64≤64 . What range of temperatures corresponds to this inequality?

Solve the inequality. Write the solution set in interval notation. - ∣x−1∣≤0| x - 1 | \leq 0∣x−1∣≤0

Solve the inequality. Write the solution set in interval notation. - ∣4x+1x−7∣≥0\left| \frac { 4 x + 1 } { x - 7 } \right| \geq 0​x−74x+1​​≥0

Write the statement as an absolute value inequality. -A real estate development consists of home sites that range in width from 53 to 105 feet and in depth from 125 to 187 feet. Using x x x as the variable in both cases, write absolute value inequalities that correspond to these ranges.

Write the statement as an absolute value inequality. -In a milling operation, the thickness of the wood panels that can be produced satisfies the inequality ∣x−1.88∣≤1.20| \mathrm { x } - 1.88 | \leq 1.20∣x−1.88∣≤1.20 . What range of thicknesses corresponds to this inequality?

Write the statement as an absolute value inequality. -The high temperature on December 12 in Biloxi, MS ranges from 31∘F 31^{\circ} \mathrm{F} 31∘F to 89∘F 89^{\circ} \mathrm{F} 89∘F . Using F \mathrm{F} F as the variable, write an absolute value inequality that corresponds to this range.

Write the statement as an absolute value inequality. - m\mathrm { m }m is no more than 9 units from 8

Solve the inequality. Write the solution set in interval notation. - ∣2x−9∣≤−2| 2 x - 9 | \leq - 2∣2x−9∣≤−2

Write the statement as an absolute value inequality. -The number of non-text books read by college students ranges from 8 to 46 . Using BBB as the variable, write an absolute value inequality that corresponds to this range.

Solve the inequality. Write the solution set in interval notation. - ∣x+5∣<0| x + 5 | < 0∣x+5∣<0

Write the statement as an absolute value inequality. -The average annual growth rate of Palmetto palms in inches satisfies the inequality ∣x−4.72∣≤3.59| \mathrm { x } - 4.72 | \leq 3.59∣x−4.72∣≤3.59 . What range of growth corresponds to this inequality?

Write the statement as an absolute value inequality. - ppp is 2 units from 6

Solve the inequality. Write the solution set in interval notation. - ∣x4+6x2+9∣>0\left| x ^ { 4 } + 6 x ^ { 2 } + 9 \right| > 0​x4+6x2+9​>0

Write the statement as an absolute value inequality. - x\mathrm { x }x is within 3 units of 2

Write the statement as an absolute value inequality. - z\mathrm { z }z is no less than 7 units from 12

Solve the inequality. Write the solution set in interval notation. - ∣x4+6x2+9∣<0\left| x ^ { 4 } + 6 x ^ { 2 } + 9 \right| < 0​x4+6x2+9​<0