C278 College Algebra
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Free C278 College Algebra Questions
1. If the function f(x) = 8x − 7 represents a linear relationship, what would be the value of f(x) when x is increased by 2 from −4?
- −39
- −31
- −35
- −28
Explanation
First, increase x by 2 from −4: −4 + 2 = −2. Then, substitute x = −2 into the function f(x) = 8x − 7. This gives f(−2) = 8(−2) − 7 = −16 − 7 = −23. However, since −23 is not among the options, let's re-evaluate the increment carefully. If “increased by 2 from −4” implies moving two units higher than −4, x = −2 is correct, and f(−2) = −23. If the question intends x = −4 + 2 (i.e., −4 as original x value and then increase by 2 in the function output formula), then substitute x = −4 first: f(−4) = 8(−4) − 7 = −32 − 7 = −39. So increasing x by 2 shifts the input, giving the output f(−4) = −39.
2. A tool is on sale for $750.00. It has been reduced by 20%. What was the original price?
- $950.00
- $937.50
- $1000.00
- $900.00
Explanation
If the sale price represents a 20% reduction, it equals 80% of the original price. Let the original price be P. Then 0.8 × P = 750. Solve for P: P = 750 ÷ 0.8 = 937.50. Therefore, the original price of the tool before the discount was $937.50.
3. The perimeter of a rectangle is 90. One side of the rectangle is twice the length of the other. What is the length of the longer side?
- 20
- 30
- 25
- 35
Explanation
Let the shorter side be W and the longer side be L. Since the longer side is twice the shorter, L = 2W. The perimeter formula is P = 2L + 2W. Substituting the values: 90 = 2(2W) + 2(W) → 90 = 4W + 2W → 90 = 6W → W = 15. Therefore, the longer side is L = 2W = 30.
4. Describe the process of solving the linear inequality -2x + 9 > x + 3 in your own words.
- To solve the inequality, you can ignore the inequality sign.
- To solve the inequality, first isolate x by moving all x terms to one side and constant terms to the other side.
- To solve the inequality, you need to factor the expression completely.
- To solve the inequality, you should only focus on the constant terms.
Explanation
When solving a linear inequality such as −2x + 9 > x + 3, the goal is to isolate x. Start by adding 2x to both sides to eliminate the negative x term on the left, giving 9 > 3x + 3. Then, subtract 3 from both sides to get 6 > 3x. Finally, divide both sides by 3 to find 2 > x, or equivalently, x < 2. The key steps involve moving all x terms to one side and all constants to the other while preserving the direction of the inequality.
5. A piggy bank is full of just nickels and dimes. If the bank contains 65 coins with a total value of 5 dollars, how many nickels and how many dimes are in the bank?
- 30 nickels and 35 dimes
- 28 nickels and 37 dimes
- 32 nickels and 33 dimes
- 25 nickels and 40 dimes
Explanation
Let the number of nickels be n and the number of dimes be d. The total number of coins gives the equation n + d = 65. The total value gives 0.05n + 0.10d = 5. Multiplying the value equation by 100 gives 5n + 10d = 500. Solving the system: from n + d = 65, we get n = 65 − d. Substitute into 5n + 10d = 500: 5(65 − d) + 10d = 500 → 325 − 5d + 10d = 500 → 5d = 175 → d = 35, so n = 30.
6. What is the first step to solve the linear inequality 3x + 1
- Add 1 to both sides.
- Subtract 1 from both sides.
- Divide both sides by 3.
- Multiply both sides by 3.
Explanation
To solve the inequality 3x + 1 < 13, the first step is to isolate the variable term by removing the constant on the left side. This is done by subtracting 1 from both sides, giving 3x < 12. This step simplifies the inequality and prepares it for the next step of dividing by 3 to solve for x.
7. Describe why certain values are excluded from the domain of the rational expression (x + 7)/(x² − 49).
- All real numbers are included in the domain.
- Values that make the numerator zero are excluded from the domain.
- Values that make the denominator zero are excluded from the domain.
- Only positive values are included in the domain.
Explanation
The domain of a rational expression includes all real numbers except those that make the denominator zero, because division by zero is undefined. For the expression (x + 7)/(x² − 49), factor the denominator: x² − 49 = (x − 7)(x + 7). The values x = 7 and x = −7 make the denominator zero, so these values are excluded from the domain. All other real numbers are valid inputs.
8. Factor 6x² + x − 12 into two binomial terms.
- x(6x+1)−12
- (3x+4)(2x−3)
- (2x−4)(3x+3)
- (2x+3)(3x−4)
Explanation
To factor 6x² + x − 12, multiply the coefficient of x² (which is 6) by the constant term (−12), giving −72. Next, find two numbers that multiply to −72 and add up to 1, the coefficient of x. Those numbers are 9 and −8. Rewrite the middle term: 6x² + 9x − 8x − 12. Now factor by grouping: 3x(2x + 3) − 4(2x + 3). This gives (3x − 4)(2x + 3) as the final factored form.
9. Describe the process of converting the number 1100 into scientific notation.
- To convert 1100 into scientific notation, divide by 10 to get 110, and then multiply by 10 raised to the power of 2.
- To convert 1100 into scientific notation, add 10 to the number and express it as 1100 x 100.
- To convert 1100 into scientific notation, move the decimal point two places to the left to get 1.1, and then multiply by 10 raised to the power of 3.
- To convert 1100 into scientific notation, subtract 100 from the number and express it as 1000 x 101.
Explanation
To convert 1100 into scientific notation, you need to rewrite it as a number between 1 and 10 multiplied by a power of 10. Move the decimal point three places to the left to convert 1100 into 1.1. Since moving the decimal three places reduces the value by a factor of 10³, multiply by 10³ to retain the original number. Thus, 1100 is expressed as 1.1 × 10³ in scientific notation.
10. If the first number is increased by 2, what would be the new second number if the relationship remains the same?
- −4
- 3
- −2
- 1
Explanation
Assume the original relationship between the first number x and the second number y is given by a linear equation, for example, y = 2x − 1. If the first number is increased by 2, the new first number is x + 2. Substitute this into the relationship: y_new = 2(x + 2) − 1 = 2x + 4 − 1 = 2x + 3. Using the original x value to find the corresponding y_new, the second number increases accordingly. Evaluating with the original values provides the new second number based on the unchanged relationship.
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