Let’s solve each question step by step.
Question 1: Solve the Compound Inequality
The compound inequality is: \[ 6c + 8 \leq -10 \quad \text{or} \quad 8c + 12 > 20 \]
Step 1: Solve the first inequality
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Start with: \[ 6c + 8 \leq -10 \]
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Subtract 8 from both sides: \[ 6c \leq -10 - 8 \] \[ 6c \leq -18 \]
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Divide both sides by 6: \[ c \leq -3 \]
Step 2: Solve the second inequality
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Start with: \[ 8c + 12 > 20 \]
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Subtract 12 from both sides: \[ 8c > 20 - 12 \] \[ 8c > 8 \]
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Divide both sides by 8: \[ c > 1 \]
Step 3: Combine the solutions
The solutions to the compound inequality are: \[ c \leq -3 \quad \text{or} \quad c > 1 \]
Question 2: Simplify the Expression
The expression to simplify is: \[ 22x^{-1}y^{-89} - 1x^{0}y^{-8} \]
Step 1: Reduce \( x^0 \)
Recall that \( x^0 = 1 \). Thus: \[ 1x^{0}y^{-8} = y^{-8} \]
Step 2: Rewrite the expression
Now the expression looks like this: \[ 22x^{-1}y^{-89} - y^{-8} \]
Step 3: Rewrite using positive exponents
To express everything using positive exponents:
- For \( x^{-1} \), rewrite as \( \frac{1}{x} \).
- For \( y^{-89} \), rewrite as \( \frac{1}{y^{89}} \).
- For \( y^{-8} \), rewrite as \( \frac{1}{y^8} \).
The expression can be rewritten as: \[ 22 \cdot \frac{y^{-89}}{x} - \frac{1}{y^8} = \frac{22}{xy^{89}} - \frac{1}{y^8} \]
Step 4: Common Denominator
To combine these fractions, we can find a common denominator: \[ xy^{89} \text{ and } y^8 \]
The least common denominator (LCD) of \( xy^{89} \) and \( y^8 \) is \( xy^{89} \).
Step 5: Adjust the second term
Rewrite \( y^{-8} \) so that it has the same denominator: \[ \frac{1}{y^8} = \frac{x \cdot y^{81}}{xy^{89}} \]
Now we express it: \[ \frac{22}{xy^{89}} - \frac{xy^{81}}{xy^{89}} \]
Combine the fractions: \[ \frac{22 - xy^{81}}{xy^{89}} \]
Final Answers
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The solutions to the compound inequality are: \[ c \leq -3 \quad \text{or} \quad c > 1 \]
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The simplified expression is: \[ \frac{22 - xy^{81}}{xy^{89}} \]