Hey I need help on the test Equations, Expressions, and inequalities unit test part two. Please show all the work! I really suck at math and would really appreciate the help.
1st Question-
Evaluate the expression for the given value of the variable.
|-4b-8| + |-1-b^2|+2b^3; b=-2
2nd Question -
Enter your answer and show all the steps that you use to solve this problem in the space provided.
Solve the equation. Check for extraneous solutions.
9|9-8x|=2x+3
1st Question:
Substitute b=-2 into the expression |-4b-8| + |-1-b^2|+2b^3:
|-4(-2)-8| + |-1-(-2)^2|+2(-2)^3 = |8-8| + |-1-4| -16 = 0+5-16 = -11
Answer: -11
2nd Question:
First, we need to simplify the expression inside the absolute value:
9|9-8x| = 2x+3
|9-8x| = (2x+3)/9
Then, we have two cases:
1) 9-8x = (2x+3)/9
Multiplying both sides by 9, we have:
81 - 72x = 2x+3
Solving for x, we get:
74x = 78
x = 39/37
However, we need to check if this solution is extraneous by substituting it back into the original equation:
9|9-8(39/37)| = 2(39/37)+3
9|(315/37)| = (78/37)+3
2835/37 = 231/37
This is not true, so x = 39/37 is an extraneous solution and we discard it.
2) 9-8x = -(2x+3)/9
Multiplying both sides by 9, we have:
81 - 72x = -(2x+3)
Solving for x, we get:
x = -87/74
Again, we need to check if this solution is extraneous:
9|9-8(-87/74)| = 2(-87/74)+3
9|(903/37)| = (-171/37)
This is not true, so x = -87/74 is also an extraneous solution and we discard it.
Answer: There are no solutions to the equation 9|9-8x|=2x+3.
Sure, I'd be happy to help you step-by-step with these math problems.
1st Question:
We are given the expression |-4b-8| + |-1-b^2| + 2b^3 and we need to evaluate it for b = -2.
Let's substitute -2 into the expression:
|-4(-2)-8| + |-1-(-2)^2| + 2(-2)^3
|-4*(-2)-8| + |-1-4| + 2*(-8)
|8-8| + |-1-4| - 16
|0| + |-5| - 16
0 + 5 - 16
5 - 16
-11
Therefore, when b = -2, the value of the expression is -11.
2nd Question:
We have the equation 9|9-8x| = 2x + 3. Let's solve it step-by-step.
Step 1: Remove the absolute value by considering both positive and negative cases.
Case 1: (9-8x) >= 0 (since the expression inside the absolute value should be non-negative)
9 - 8x >= 0
Step 2: Solve the inequality from Case 1.
-8x >= -9
Divide both sides by -8 and remember to reverse the inequality sign:
x <= 9/8
Case 2: (9-8x) < 0
9 - 8x < 0
Step 3: Solve the inequality from Case 2.
-8x < -9
Divide both sides by -8 and remember to reverse the inequality sign:
x > 9/8
Step 4: Combine the solutions from Step 2 and Step 3.
The solution for x is x <= 9/8 or x > 9/8.
Step 5: Check for extraneous solutions by substituting the solutions into the original equation.
For x = 9/8:
9|9-8(9/8)| = 2(9/8) + 3
9|9-9| = 18/8 + 3
9|0| = 9/4 + 3
0 = 9/4 + 12/4
0 = 21/4
Since 0 does not equal 21/4, x = 9/8 is not a solution.
Therefore, the only valid solution is x > 9/8.