Question

inequalities puzzle
questions

1. 88 + r > -16
2. 9 + 1 - 4 > 2
3. 93 < -3 + 6(2x + 6)
4. -98 < 8( -4 + x)
5. 8(-2k - 1) + 5(2k - 4) < -46
6. 4(2v + 8) + 8(6v - 4) < -80

answers

1. x > -11

2.

1. x > 5

4.
5.
6.
questions

1. 1 - 4p > 1 + 5p
2. -15 + 5n > n - 7
3. 5(-4n + 1) < -7n - 34
4. 8(1 - 2x) < -13 + 5x
5. -(4x + 1) < -7x - 34
6. -4(5 + 7x) - x < -6(1 + 6x)

answers

1. x
2. n > 2

9.
10.

1. x < -11
2. x < 2

directions: solve each inequality. your answer should be the answer, copy, and then the number (ex: 1 or 4). if you are correct a piece of the puzzle will show. if you are incorrect the puzzle will not change.

Answer:

To solve these inequalities, we'll go through each one step by step. Let's start with the first few:

Question 1: \( 8r + r > -18 \)

Step 1: Combine like terms

\( 9r > -18 \)

Step 2: Divide both sides by 9

\( r > \frac{-18}{9} \)
\( r > -2 \) Wait, but the answer given is \( x > -11 \). Maybe I misread the variable. Let's check again. Oh, maybe it's \( 8x + x > -18 \)? Let's assume that.

Step 1: Combine like terms

\( 9x > -18 \)

Step 2: Divide by 9

\( x > \frac{-18}{9} \)
\( x > -2 \). Hmm, not matching. Maybe the original is \( 8 + r > -18 \)? Let's try that.

Step 1: Subtract 8 from both sides

\( r > -18 - 8 \)
\( r > -26 \). Still not. Maybe a typo, but let's move to the next.

Question 2: \( 9 + 1 - 4 > 2 \)

Wait, that's \( 6 > 2 \), which is always true. Maybe it's \( 9 + n - 4 > 2 \)? Let's assume \( 9 + n - 4 > 2 \)

Step 1: Simplify left side

\( 5 + n > 2 \)

Step 2: Subtract 5

\( n > 2 - 5 \)
\( n > -3 \). Not matching the answer format.

Question 3: \( 93 < -3 + 6(2x + 6) \)

Step 1: Distribute the 6

\( 93 < -3 + 12x + 36 \)

Step 2: Combine like terms

\( 93 < 12x + 33 \)

Step 3: Subtract 33

\( 93 - 33 < 12x \)
\( 60 < 12x \)

Step 4: Divide by 12

\( \frac{60}{12} < x \)
\( 5 < x \) or \( x > 5 \). Which matches the given answer \( x > 5 \). Great, that works.

Question 4: \( -98 < 8(-4 + x) \)

Step 1: Distribute the 8

\( -98 < -32 + 8x \)

Step 2: Add 32 to both sides

\( -98 + 32 < 8x \)
\( -66 < 8x \)

Step 3: Divide by 8

\( x > \frac{-66}{8} \)
\( x > -8.25 \). Not matching the answer format.

Question 5: \( 8(-2x - 1) + 5(2x - 4) < -46 \)

Step 1: Distribute both 8 and 5

\( -16x - 8 + 10x - 20 < -46 \)

Step 2: Combine like terms

\( -6x - 28 < -46 \)

Step 3: Add 28 to both sides

\( -6x < -46 + 28 \)
\( -6x < -18 \)

Step 4: Divide by -6 (remember to flip inequality)

\( x > \frac{-18}{-6} \)
\( x > 3 \). Not matching.

Question 6: \( 4(2x + 8) + 8(6x - 4) < -80 \)

Step 1: Distribute 4 and 8

\( 8x + 32 + 48x - 32 < -80 \)

Step 2: Combine like terms

\( 56x < -80 \)

Step 3: Divide by 56

\( x < \frac{-80}{56} \)
\( x < \frac{-10}{7} \approx -1.43 \). Not matching.

Question 7: \( 1 - 4p > 1 + 5p \)

Step 1: Subtract 1 from both sides

\( -4p > 5p \)

Step 2: Subtract 5p

\( -9p > 0 \)

Step 3: Divide by -9 (flip inequality)

\( p < 0 \)

Question 8: \( -15 + 5n > n - 7 \)

Step 1: Subtract n from both sides

\( -15 + 4n > -7 \)

Step 2: Add 15

\( 4n > 8 \)

Step 3: Divide by 4

\( n > 2 \)

Question 9: \( 5(-4n + 1) < -7n - 34 \)

Step 1: Distribute 5

\( -20n + 5 < -7n - 34 \)

Step 2: Add 20n to both sides

\( 5 < 13n - 34 \)

Step 3: Add 34

\( 39 < 13n \)

Step 4: Divide by 13

\( 3 < n \)
\( n > 3 \)

Question 10: \( 8(1 - 2x) < -13 + 5x \)

Step 1: Distribute 8

\( 8 - 16x < -13 + 5x \)

Step 2: Add 16x to both sides

\( 8 < 21x - 13 \)

Step 3: Add 13

\( 21 < 21x \)

Step 4: Divide by 21

\( 1 < x \)
\( x > 1 \)

Question 11: \( -(4x + 1) < -7x - 34 \)

Step 1: Distribute the negative sign

\( -4x - 1 < -7x - 34 \)

Step 2: Add 7x to both sides

\( 3x - 1 < -34 \)

Step 3: Add 1

\( 3x < -33 \)

Step 4: Divide by 3

\( x < -11 \)

Question 12: \( -4(5 + 7x) - x < -6(1 + 6x) \)

Step 1: Distribute -4 and -6

\( -20 - 28x - x < -6 - 36x \)

Step 2: Combine like terms

\( -20 - 29x < -6 - 36x \)

Step 3: Add 36x to both sides

\( -20 + 7x < -6 \)

Step 4: Add 20

\( 7x < 14 \)

Step 5: Divide by 7

\( x < 2 \)

Now let's match the answers:

• Question 3: \( 93 < -3 + 6(2x + 6) \) gives \( x > 5 \) (matches the answer)
• Question 11: \( -(4x + 1) < -7x - 34 \) gives \( x < -11 \) (matches the answer)
• Question 12: \( -4(5 + 7x) - x < -6(1 + 6x) \) gives \( x < 2 \) (matches the answer)
• Question 8: \( -15 + 5n > n - 7 \) gives \( n > 2 \) (matches the answer)

Let's verify Question 11 again:

\( -(4x + 1) < -7x - 34 \)
\( -4x - 1 < -7x - 34 \)
Add \( 7x \) to both sides: \( 3x - 1 < -34 \)
Add 1: \( 3x < -33 \)
Divide by 3: \( x < -11 \). Correct.

Question 12:

\( -4(5 + 7x) - x < -6(1 + 6x) \)
\( -20 - 28x - x < -6 - 36x \)
\( -20 - 29x < -6 - 36x \)
Add \( 36x \): \( -20 + 7x < -6 \)
Add 20: \( 7x < 14 \)
Divide by 7: \( x < 2 \). Correct.

Question 8:

\( -15 + 5n > n - 7 \)
Subtract \( n \): \( -15 + 4n > -7 \)
Add 15: \( 4n > 8 \)
Divide by 4: \( n > 2 \). Correct.

Question 3:

\( 93 < -3 + 6(2x + 6) \)
\( 93 < -3 + 12x + 36 \)
\( 93 < 12x + 33 \)
Subtract 33: \( 60 < 12x \)
Divide by 12: \( 5 < x \) or \( x > 5 \). Correct.

So these are the solutions for the given inequalities.