Question 1

Solve the inequality. Write the solution set in interval notation.
-$$| - 2 x + 3 | > - 2$$
A) $\varnothing$
B) $\left( \frac { 1 } { 2 } , \frac { 5 } { 2 } \right)$
C) $\left( - \infty , \frac { 5 } { 2 } \right)$
D) $( - \infty , \infty )$

Accepted Answer

The inequality $| - 2 x + 3 | > - 2$ is always true because the absolute value of any real number is always non-negative, and thus always greater than any negative number. Therefore, the solution set includes all real numbers, represented by $( - \infty , \infty )$.

Question 2

Write the statement as an absolute value inequality.
-$x$ is within 10 units of 9
A) $| x - 10 | < 9$
B) $| x - 9 | < 10$
C) $| x - 9 | \leq 10$
D) $| 10 - x | < 9$

Accepted Answer

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

Question 3

The temperatures in parts of Antarctica in degrees Celsius roughly satisfy the inequality $| C + 92 | \leq 53$. What range of temperatures corresponds to this inequality?
A) $\left[ - 145 ^ { \circ } \mathrm { C } , - 39 ^ { \circ } \mathrm { C } \right]$
B) $\left[ - 145 ^ { \circ } \mathrm { C } , 145 ^ { \circ } \mathrm { C } \right]$
C) $\left[ - 39 ^ { \circ } \mathrm { C } , 145 ^ { \circ } \mathrm { C } \right]$
D) $\left[ - 39 ^ { \circ } \mathrm { C } , 39 ^ { \circ } \mathrm { C } \right]$

Accepted Answer

To solve the inequality $| C + 92 | \leq 53$, we break it into two cases: $C + 92 \leq 53$ and $-(C + 92) \leq 53$. Solving the first gives $C \leq -39$, and solving the second gives $C \geq -145$. Thus, the range of temperatures is $[-145, -39]$.

Question 4

Solve the inequality. Write the solution set in interval notation.
-| $$3 x - 9 \mid \leq - 6$$
A) $\varnothing$
B) $( 5,1 )$
C) $( - \infty , 5 ) \cup ( 1 , \infty )$
D) $\left( - \infty , - \frac { 1 } { 3 } \right) \cup \left( - \frac { 13 } { 3 } , \infty \right)$

Accepted Answer

The inequality $|3x - 9| \leq -6$ has no solution because the absolute value of any real number is always non-negative, and it cannot be less than a negative number. Thus, the solution set is empty, denoted by $\varnothing$.

Question 5

Solve the inequality. Write the solution set in interval notation.
-$$| x + 3 | < 0$$
A) $\{ 3 \}$
B) $( - \infty , - 3 )$
C) $\varnothing$
D) $\{ - 3 \}$

Accepted Answer

The absolute value of any real number is always non-negative, meaning it is either positive or zero. Therefore, the inequality $| x + 3 | < 0$ has no solution, as the absolute value cannot be less than zero.

Question 6

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, $x^4 + 6x^2 + 9$, is a sum of squares and is always non-negative. Specifically, it can be factored as $(x^2 + 3)^2$, which is always greater than or equal to 0. Since the absolute value of a non-negative number cannot be less than 0, there are no solutions to the inequality. Therefore, the solution set is the empty set, denoted by $\varnothing$.

Question 7

Write the statement as an absolute value inequality.
-p is 5 units from 8 
A) $| p - 5 | = 8$
B) $| p - 8 | < 5$
C) $| \mathrm { p } | = 3$
D) $| p - 8 | = 5$

Accepted Answer

The absolute value inequality $| p - 8 | = 5$ represents that p is exactly 5 units away from 8.

Question 8

Solve the inequality. Write the solution set in interval notation.
-$$\left| \frac { 4 x + 1 } { x - 6 } \right| \geq 0$$
A) $( - 6,6 )$
B) $\left( - \infty , - \frac { 1 } { 4 } \right) \cup ( 6 , \infty )$
C) $\left( - \frac { 1 } { 4 } , 6 \right)$
D) $( - \infty , 6 ) \cup ( 6 , \infty )$

Accepted Answer

The absolute value of any real number or algebraic expression is always greater than or equal to zero. The only restriction comes from the denominator of the fraction, which cannot be zero. Thus, $x$ cannot be 6. The solution set includes all real numbers except 6, represented by $( - \infty , 6 ) \cup ( 6 , \infty )$.

Question 9

Write the statement as an absolute value inequality.
-m is no more than 7 units from 19 
A) $| m - 19 | > 7$
B) $| m - 19 | \leq 7$
C) $| m - 19 | < 7$
D) $| m - 7 | < 19$

Accepted Answer

The statement "m is no more than 7 units from 19" means the distance between m and 19 is less than or equal to 7, which is represented by $| m - 19 | \leq 7$.

Question 10

The number of non-text books read by college students ranges from 6 to 44 . Using B as the variable, write an absolute value inequality that corresponds to this range.
A) $| B - 38 | \leq 6$
B) $| \mathrm { B } - 19 | \leq 25$
C) $| B - 6 | \leq 38$
D) $| B - 25 | \leq 19$

Accepted Answer

The correct absolute value inequality is $| B - 25 | \leq 19$. This is because the midpoint of the range from 6 to 44 is 25 (the average of 6 and 44), and the distance from this midpoint to either endpoint of the range is 19.

Question 11

The average annual growth rate of Cyprus trees in inches satisfies the inequality |x - 3.25| 2.69. What range of growth corresponds to this inequality?
A) [0.56, 11.88]
B) [0.56, 5.94]
C) [2.69, 3.25]
D) [0.28, 5.94]

Accepted Answer

The answer of The average annual growth rate of Cyprus...

Question 12

Solve the inequality. Write the solution set in interval notation.
-$$| - 3 x - 9 | \geq - 3$$
A) $\varnothing$
B) $( - \infty , - 2 )$
C) $( - \infty , \infty )$
D) $( - 4 , - 2 )$

Accepted Answer

The inequality $| - 3 x - 9 | \geq - 3$ is always true because the absolute value of any real number is always non-negative, and thus it is always greater than or equal to any negative number. Therefore, the solution set includes all real numbers, which is represented by the interval notation $( - \infty , \infty )$.

Question 13

Write the statement as an absolute value inequality.
-z is no less than 6 units from 17 
A) $| z - 17 | \geq 6$
B) $| z - 17 | > 6$
C) $| z - 17 | \leq 6$
D) $| z - 6 | > 17$

Accepted Answer

The statement "z is no less than 6 units from 17" means the distance between z and 17 is at least 6, which is represented by $| z - 17 | \geq 6$.

Question 14

A real estate development consists of home sites that range in width from 59 to 115 feet and in depth from 135 to 191 feet. Using $x$ as the variable in both cases, write absolute value inequalities that correspond to these ranges.
A) $| \mathrm { B } - 56 | \leq 59 , | x - 56 | \leq 135$
B) $| \mathrm { B } - 59 | \leq 56 , | \mathrm { x } - 135 | \leq 56$
C) $| x - 87 | \leq 28 , | x - 163 | \leq 28$
D) $| x - 28 | \leq 87 , | x - 28 | \leq 163$

Accepted Answer

The correct absolute value inequalities are found by determining the midpoint of each range and then calculating the maximum deviation from that midpoint. For width, the midpoint between 59 and 115 is 87, and the range is 56, so the deviation is 28 (half of 56). For depth, the midpoint between 135 and 191 is 163, and the range is 56, so the deviation is also 28. Therefore, $| x - 87 | \leq 28$ and $| x - 163 | \leq 28$ correctly represent the given ranges.

Question 15

Solve the inequality. Write the solution set in interval notation.
-$$| x + 2 | \leq 0$$
A) $\{ - 2 \}$
B) $( - \infty , - 2 )$
C) $\{ 2 \}$
D) $\varnothing$

Accepted Answer

The absolute value of any real number is always greater than or equal to zero. The only way |x + 2| can equal 0 is if x + 2 = 0, which means x = -2. Therefore, the solution set is the single value {-2}.

Question 16

In a milling operation, the thickness of the metal bars that can be produced satisfies the inequality $| x - 1.97 | \leq 1.38$. What range of thicknesses corresponds to this inequality?
A) $[ 1.38,1.97 ]$
B) $[ 0.59,3.35 ]$
C) $[ 0.59,6.7 ]$
D) $[ 0.3,3.35 ]$

Accepted Answer

The inequality $| x - 1.97 | \leq 1.38$ can be solved by splitting it into two cases: $x - 1.97 \leq 1.38$ and $-(x - 1.97) \leq 1.38$, which simplifies to $x \leq 3.35$ and $x \geq 0.59$, respectively. Therefore, the range of thicknesses is $[0.59, 3.35]$.

Question 17

Solve the inequality. Write the solution set in interval notation.
-$$\left| x ^ { 4 } + 8 x ^ { 2 } + 16 \right| > 0$$
A) $( - 4,4 )$
B) $\varnothing$
C) $( - \infty , - 4 ) \cup ( 4 , \infty )$
D) $( - \infty , \infty )$

Accepted Answer

The expression inside the absolute value, $x^4 + 8x^2 + 16$, can be factored as $(x^2 + 4)^2$, which is always non-negative. Since it's squared, it's always positive except possibly at the roots of $x^2 + 4 = 0$. However, $x^2 + 4 = 0$ has no real solutions because $x^2 = -4$ is not possible for real numbers. Therefore, the original expression is always positive for all real x, making the solution set all real numbers, $(-\infty, \infty)$.

Question 18

The high temperature on December 12 in Hilton Head, $\mathrm { SC }$ ranges from $33 ^ { \circ } \mathrm { F }$ to $89 ^ { \circ } \mathrm { F }$. Using $\mathrm { F }$ as the variable, write an absolute value inequality that corresponds to this range.
A) $| F - 28 | \leq 61$
B) $| F - 33 | \ll 56$
C) $| \mathrm { F } - 56 | \leq 33$
D) $| F - 61 | \leq 28$

Accepted Answer

The absolute value inequality $| F - 61 | \leq 28$ represents the range from $33 ^ { \circ } \mathrm { F }$ to $89 ^ { \circ } \mathrm { F }$ because the midpoint of the range is 61, and the distance from the midpoint to either endpoint is 28.

Quiz 2: Equations and Inequalities

Solve the inequality. Write the solution set in interval notation. - $| - 2 x + 3 | > - 2$

Write the statement as an absolute value inequality. - $x$ is within 10 units of 9

The temperatures in parts of Antarctica in degrees Celsius roughly satisfy the inequality $| C + 92 | \leq 53$ . What range of temperatures corresponds to this inequality?

Solve the inequality. Write the solution set in interval notation. -| $3 x - 9 \mid \leq - 6$

Solve the inequality. Write the solution set in interval notation. - $| x + 3 | < 0$

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. -p is 5 units from 8

Solve the inequality. Write the solution set in interval notation. - $\left| \frac { 4 x + 1 } { x - 6 } \right| \geq 0$

Write the statement as an absolute value inequality. -m is no more than 7 units from 19

The number of non-text books read by college students ranges from 6 to 44 . Using B as the variable, write an absolute value inequality that corresponds to this range.

The average annual growth rate of Cyprus trees in inches satisfies the inequality |x - 3.25| 2.69. What range of growth corresponds to this inequality?

Solve the inequality. Write the solution set in interval notation. - $| - 3 x - 9 | \geq - 3$

Write the statement as an absolute value inequality. -z is no less than 6 units from 17

A real estate development consists of home sites that range in width from 59 to 115 feet and in depth from 135 to 191 feet. Using $x$ as the variable in both cases, write absolute value inequalities that correspond to these ranges.

Solve the inequality. Write the solution set in interval notation. - $| x + 2 | \leq 0$

In a milling operation, the thickness of the metal bars that can be produced satisfies the inequality $| x - 1.97 | \leq 1.38$ . What range of thicknesses corresponds to this inequality?

Solve the inequality. Write the solution set in interval notation. - $\left| x ^ { 4 } + 8 x ^ { 2 } + 16 \right| > 0$

The high temperature on December 12 in Hilton Head, $\mathrm { SC }$ ranges from $33 ^ { \circ } \mathrm { F }$ to $89 ^ { \circ } \mathrm { F }$ . Using $\mathrm { F }$ as the variable, write an absolute value inequality that corresponds to this range.

Quiz 2: Equations and Inequalities

Solve the inequality. Write the solution set in interval notation. - ∣−2x+3∣>−2| - 2 x + 3 | > - 2∣−2x+3∣>−2

Write the statement as an absolute value inequality. - xxx is within 10 units of 9

The temperatures in parts of Antarctica in degrees Celsius roughly satisfy the inequality ∣C+92∣≤53| C + 92 | \leq 53∣C+92∣≤53 . What range of temperatures corresponds to this inequality?

Solve the inequality. Write the solution set in interval notation. -| 3x−9∣≤−63 x - 9 \mid \leq - 63x−9∣≤−6

Solve the inequality. Write the solution set in interval notation. - ∣x+3∣<0| x + 3 | < 0∣x+3∣<0

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