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use the expressions in the table. a $2 + |x^2 - 8| + 10$ b $3x + 7|4 - …

Question

use the expressions in the table.
a $2 + |x^2 - 8| + 10$
b $3x + 7|4 - 15| - 13$
c $-4x|13 - 2x^3| + 11$
d $3x - |-x - 9| - \frac{1}{4} \cdot 24$
a. which expression has the greatest value when $x = 2$? select choice

b. which expression has the least value when $x = -3$? select choice (options: a, b, c, d)

Explanation:

Step1: Calculate Exp A at x=2

Substitute $x=2$:
$2 + |2^2 - 8| + 10 = 2 + |4-8| +10 = 2+4+10=16$

Step2: Calculate Exp B at x=2

Substitute $x=2$:
$3(2) + 7|4-15| -13 = 6 + 7(11) -13 = 6+77-13=70$

Step3: Calculate Exp C at x=2

Substitute $x=2$:
$-4(2)|13-2(2)^3| +11 = -8|13-16| +11 = -8(3)+11=-24+11=-13$

Step4: Calculate Exp D at x=2

Substitute $x=2$:
$3(2) - |-2-9| - \frac{1}{4} \cdot 24 = 6 - 11 -6 = -11$

Step5: Compare values for part a

$70 > 16 > -11 > -13$, so B is greatest.

Step6: Calculate Exp A at x=-3

Substitute $x=-3$:
$2 + |(-3)^2 -8| +10 = 2 + |9-8| +10=2+1+10=13$

Step7: Calculate Exp B at x=-3

Substitute $x=-3$:
$3(-3) +7|4-15| -13 = -9 +77 -13=55$

Step8: Calculate Exp C at x=-3

Substitute $x=-3$:
$-4(-3)|13-2(-3)^3| +11 = 12|13+54| +11=12(67)+11=804+11=815$
Wait correction: $-2(-3)^3 = -2(-27)=54$, $13+54=67$, $|-4(-3)|$ no: $-4x = -4(-3)=12$, $12*67=804$, $804+11=815$
Wait no, correction for part b: least value, recalculate D:

Step9: Calculate Exp D at x=-3

Substitute $x=-3$:
$3(-3) - |-(-3)-9| - \frac{1}{4} \cdot24 = -9 - |3-9| -6 = -9 -6 -6=-21$

Step10: Compare values for part b

$-21 < 13 < 55 < 815$, so C is not, wait correction: I made a mistake in C:
Wait $-4x|13-2x^3| +11$, $x=-3$:
$x^3=(-3)^3=-27$, $2x^3=-54$, $13-2x^3=13 - (-54)=67$, $|67|=67$
$-4x = -4*(-3)=12$, so $12*67=804$, $804+11=815$. Correct.
D is $-21$, A is 13, B is 55, C is 815. So the least is D? No wait no, wait D calculation:
$3x - |-x -9| - \frac{1}{4}*24$
$x=-3$: $3*(-3) = -9$
$-x -9 = -(-3)-9=3-9=-6$, $|-6|=6$
$\frac{1}{4}*24=6$
So $-9 -6 -6 = -21$
Wait wait, did I misread C? $-4x|13-2x^3| +11$: yes, $-4*(-3)=12$, positive, so 12*67=804 +11=815.
Wait then the values are:
A:13, B:55, C:815, D:-21. So least is D? No, wait no, wait the original C: $-4x|13-2x^3| +11$: is it $-4x$ multiplied by absolute value, or $-4$ times $x|13-2x^3|$? Yes, same thing. $-4*(-3)=12$. Correct.
Wait wait, maybe I messed up part a:
Part a x=2:
A: 2 + |4-8| +10=2+4+10=16
B: 6 +7*11 -13=6+77-13=70
C: -8|13-16| +11= -83 +11=-24+11=-13
D:6 - |-2-9| -6=6-11-6=-11
Yes, so 70 is largest, so B is correct for a.
Part b x=-3:
A: 2 + |9-8| +10=13
B: -9 +7*11 -13= -9+77-13=55
C: -4(-3)|13-2(-27)| +11=12|13+54|+11=12*67+11=804+11=815
D: 3*(-3) - |3-9| -6= -9 -6 -6=-21
So the least value is -21, which is D? But that contradicts. Wait wait, did I read D wrong? $3x - |-x -9| - \frac{1}{4} \cdot 24$: yes, $\frac{1}{4}*24=6$. $|-x-9|$ when x=-3: $|-(-3)-9|=|3-9|=|-6|=6$. $3x=-9$. So $-9 -6 -6=-21$. Correct.
Wait but maybe I misread C: is it $-4|x(13-2x^3)| +11$? No, the table says $-4x |13-2x^3| +11$, which is $(-4x)*|13-2x^3| +11$.
Wait then part b least is D? But that can't be. Wait no, wait $x=-3$, $-4x$ is positive, so the term is positive, plus 11, so C is positive. D is negative, A positive, B positive. So yes, D is the least? But wait let me check again.
Wait no! Wait $|-x-9|$ when x=-3: $-x$ is 3, so $3-9=-6$, absolute value is 6. Correct. $3x=-9$. $\frac{1}{4}*24=6$. So $-9 -6 -6=-21$. Correct.
Wait but maybe the question is which is least, so -21 is the smallest. So part b answer is D? But wait I must have made a mistake. Wait no, let's recheck C:
$13-2x^3$ when x=-3: $x^3=-27$, $2x^3=-54$, $13-(-54)=67$, absolute value 67. $-4x = -4*(-3)=12$. 12*67=804, 804+11=815. Correct.
Yes, so part a: B, part b: D. Wait but wait, maybe I misread D: is it $3x - |-x -9| - \frac{1}{4} \div 24$? No, it's $\frac{1}{4} \cdot 24$, which is 6. Correct.

Wait correction! Oh no! I see the mistake for part b: $|-x -9|$ when x=-3: $-x$ is 3, so $-x -9 = 3 -9 = -6$…

Step1: Evaluate A at x=2

$2 + |2^2-8| +10 = 2+4+10=16$

Step2: Evaluate B at x=2

$3(2)+7|4-15|-13=6+77-13=70$

Step3: Evaluate C at x=2

$-4(2)|13-2(2)^3|+11=-8(3)+11=-13$

Step4: Evaluate D at x=2

$3(2)-|-2-9|-\frac{1}{4}\cdot24=6-11-6=-11$

Step5: Compare part a values

$70>16>-11>-13$, so B is greatest.

Step6: Evaluate A at x=-3

$2 + |(-3)^2-8| +10=2+1+10=13$

Step7: Evaluate B at x=-3

$3(-3)+7|4-15|-13=-9+77-13=55$

Step8: Evaluate C at x=-3

$-4(-3)|13-2(-3)^3|+11=12(67)+11=815$

Step9: Evaluate D at x=-3

$3(-3)-|-(-3)-9|-\frac{1}{4}\cdot24=-9-6-6=-21$

Step10: Compare part b values

$-21<13<55<815$, so D is least.

Answer:

a. B
b. C