Question

solve each inequality and graph its solution. put your answer in set notation.

1. $9|9r - 9| - 6 \leq 75$
2. $7|-5 - 4x| - 5 \geq 100$
3. $10|6 - b| - 3 > 67$
4. $10 + 4|-m - 7| < 18$

solve each inequality. put your answer in set notation.

1. $10|9n - 1| + 10 \geq 110$
2. $4|5 - a| - 6 \leq 22$

Explanation:

Step1: Solve Inequality 7

Given \( 9|9r - 9| - 6 \leq 75 \)
First, add 6 to both sides: \( 9|9r - 9| \leq 75 + 6 = 81 \)
Then divide by 9: \( |9r - 9| \leq \frac{81}{9} = 9 \)
This gives two inequalities: \( -9 \leq 9r - 9 \leq 9 \)
Add 9 to all parts: \( 0 \leq 9r \leq 18 \)
Divide by 9: \( 0 \leq r \leq 2 \)
In set notation: \( \{r | 0 \leq r \leq 2\} \)

Step2: Solve Inequality 8

Given \( 7|-5 - 4x| - 5 \geq 100 \)
Add 5 to both sides: \( 7|-5 - 4x| \geq 105 \)
Divide by 7: \( |-5 - 4x| \geq 15 \)
This gives two cases:
Case 1: \( -5 - 4x \geq 15 \)
Subtract -5: \( -4x \geq 20 \)
Divide by -4 (reverse inequality): \( x \leq -5 \)
Case 2: \( -5 - 4x \leq -15 \)
Subtract -5: \( -4x \leq -10 \)
Divide by -4 (reverse inequality): \( x \geq \frac{5}{2} = 2.5 \)
In set notation: \( \{x | x \leq -5 \text{ or } x \geq 2.5\} \)

Step3: Solve Inequality 9

Given \( 10|6 - b| - 3 > 67 \)
Add 3 to both sides: \( 10|6 - b| > 70 \)
Divide by 10: \( |6 - b| > 7 \)
This gives two cases:
Case 1: \( 6 - b > 7 \)
Subtract 6: \( -b > 1 \)
Multiply by -1 (reverse inequality): \( b < -1 \)
Case 2: \( 6 - b < -7 \)
Subtract 6: \( -b < -13 \)
Multiply by -1 (reverse inequality): \( b > 13 \)
In set notation: \( \{b | b < -1 \text{ or } b > 13\} \)

Step4: Solve Inequality 10

Given \( 10 + 4|-m - 7| < 18 \)
Subtract 10: \( 4|-m - 7| < 8 \)
Divide by 4: \( |-m - 7| < 2 \)
This is equivalent to \( |m + 7| < 2 \) (since \( |-m -7| = |m +7| \))
Gives \( -2 < m + 7 < 2 \)
Subtract 7: \( -9 < m < -5 \)
In set notation: \( \{m | -9 < m < -5\} \)

Step5: Solve Inequality 11

Given \( 10|9n - 1| + 10 \geq 110 \)
Subtract 10: \( 10|9n - 1| \geq 100 \)
Divide by 10: \( |9n - 1| \geq 10 \)
This gives two cases:
Case 1: \( 9n - 1 \geq 10 \)
Add 1: \( 9n \geq 11 \)
Divide by 9: \( n \geq \frac{11}{9} \approx 1.22 \)
Case 2: \( 9n - 1 \leq -10 \)
Add 1: \( 9n \leq -9 \)
Divide by 9: \( n \leq -1 \)
In set notation: \( \{n | n \leq -1 \text{ or } n \geq \frac{11}{9}\} \)

Step6: Solve Inequality 12

Given \( 4|5 - a| - 6 \leq 22 \)
Add 6 to both sides: \( 4|5 - a| \leq 28 \)
Divide by 4: \( |5 - a| \leq 7 \)
This gives two inequalities: \( -7 \leq 5 - a \leq 7 \)
Subtract 5: \( -12 \leq -a \leq 2 \)
Multiply by -1 (reverse inequalities): \( -2 \leq a \leq 12 \)
In set notation: \( \{a | -2 \leq a \leq 12\} \)

Answer:

1. \( \{r | 0 \leq r \leq 2\} \)
2. \( \{x | x \leq -5 \text{ or } x \geq 2.5\} \)
3. \( \{b | b < -1 \text{ or } b > 13\} \)
4. \( \{m | -9 < m < -5\} \)
5. \( \{n | n \leq -1 \text{ or } n \geq \frac{11}{9}\} \)
6. \( \{a | -2 \leq a \leq 12\} \)