Detailed step-by-step solutions for each question in Quiz Set 1.
Question: Which expression shows "five less than a number \(x\)"?
Solution: "Five less than" means we subtract 5 from the number \(x\). Therefore, the expression is \(x - 5\).
Correct Answer: (c) \( x - 5 \)
Question: Solve for \( n \): \( n + 3 = 10 \)
Solution: To solve for \(n\), subtract 3 from both sides of the equation: \( n + 3 - 3 = 10 - 3 \), which simplifies to \( n = 7 \).
Correct Answer: (b) \( 7 \)
Question: Which inequality represents "\(y\) is greater than or equal to two"?
Solution: "Greater than or equal to" is represented by the symbol \( \geq \). So, "\(y\) is greater than or equal to two" is written as \( y \geq 2 \).
Correct Answer: (c) \( y \geq 2 \)
Question: Calculate: \( -8 + 5 \)
Solution: When adding a positive number to a negative number, we find the difference of their absolute values and use the sign of the number with the larger absolute value. \( |-8| = 8 \) and \( |5| = 5 \). \( 8 - 5 = 3 \). Since \( -8 \) has a larger absolute value and is negative, the answer is \( -3 \).
Correct Answer: (b) \( -3 \)
Question: Simplify: \( x^2 \cdot x^3 \)
Solution: Using the exponent rule \( a^m \cdot a^n = a^{m+n} \), we add the exponents: \( x^{2+3} = x^5 \).
Correct Answer: (a) \( x^5 \)
Question: Solve for \( x \): \( 3x - 7 = 8 \)
Solution: First, add 7 to both sides: \( 3x - 7 + 7 = 8 + 7 \), which gives \( 3x = 15 \). Then, divide both sides by 3: \( \frac{3x}{3} = \frac{15}{3} \), resulting in \( x = 5 \).
Correct Answer: (b) \( x = 5 \)
Question: Which graph represents the inequality \( x < 4 \)?
Solution: The inequality \( x < 4 \) means \(x\) can be any number less than 4, but not including 4. On a number line, this is represented by an open circle at 4 (to exclude 4) and shading to the left (for all numbers less than 4).
Correct Answer: (c) Number line with open circle at 4, shading to the left
Question: Evaluate: \( \frac{3}{4} + (-\frac{1}{2}) \)
Solution: To add these fractions, we need a common denominator, which is 4. \( -\frac{1}{2} \) is equivalent to \( -\frac{2}{4} \). So, \( \frac{3}{4} + (-\frac{2}{4}) = \frac{3 - 2}{4} = \frac{1}{4} \).
Correct Answer: (c) \( \frac{1}{4} \)
Question: Simplify: \( (2^3)^2 \)
Solution: Using the exponent rule \( (a^m)^n = a^{m \cdot n} \), we multiply the exponents: \( 2^{3 \times 2} = 2^6 \).
Correct Answer: (b) \( 2^6 \)
Question: Which ordered pair is part of the function \( f(x) = 2x - 1 \)?
Solution: We test each ordered pair. For (c) \( (2, 3) \), if \( x = 2 \), \( f(2) = 2(2) - 1 = 4 - 1 = 3 \). So, \( (2, 3) \) is on the function. Let's quickly check others: (a) \( f(0) = 2(0) - 1 = -1 \neq 1 \). (b) \( f(1) = 2(1) - 1 = 1 \neq 2 \). (d) \( f(3) = 2(3) - 1 = 5 \neq 7 \).
Correct Answer: (c) \( (2, 3) \)
Question: Solve for \( x \): \( \frac{x}{2} + 3 = -1 \)
Solution: First, subtract 3 from both sides: \( \frac{x}{2} + 3 - 3 = -1 - 3 \), which gives \( \frac{x}{2} = -4 \). Then, multiply both sides by 2: \( 2 \times \frac{x}{2} = 2 \times -4 \), resulting in \( x = -8 \).
Correct Answer: (a) \( x = -8 \)
Question: Solve the inequality: \( 2x + 5 < 11 \)
Solution: Subtract 5 from both sides: \( 2x + 5 - 5 < 11 - 5 \), which simplifies to \( 2x < 6 \). Divide both sides by 2: \( \frac{2x}{2} < \frac{6}{2} \), resulting in \( x < 3 \).
Correct Answer: (a) \( x < 3 \)
Question: Evaluate: \( \frac{-6 \times 2}{3} + 4 \)
Solution: First, multiply \( -6 \times 2 = -12 \). Then divide by 3: \( \frac{-12}{3} = -4 \). Finally, add 4: \( -4 + 4 = 0 \).
Correct Answer: (c) \( 0 \)
Question: Simplify: \( \frac{a^5}{a^2} \cdot a^{-3} \)
Solution: Using exponent rules: \( \frac{a^5}{a^2} = a^{5-2} = a^3 \). Then, \( a^3 \cdot a^{-3} = a^{3 + (-3)} = a^0 \). Any non-zero number raised to the power of 0 is 1. Thus, \( a^0 = 1 \).
Correct Answer: (b) \( a^{0} \) or \( 1 \)
Question: If \( f(x) = x^2 - 3x \), find \( f(-2) \)
Solution: Substitute \( x = -2 \) into the function: \( f(-2) = (-2)^2 - 3(-2) = 4 - (-6) = 4 + 6 = 10 \).
Correct Answer: (d) \( 10 \)