Jaime’s goal is to average at least 280 miles per week for 4 weeks. If T is the total number of miles Jamie will bicycle for 4 weeks, then his goal can be represented symbolically by the inequality:^T⁄₄, or equivalently T ≥ 4(280). The total number of miles Jamie will bicycle during this time is the sum of the distances he has completed and has yet to complete. Thus T = 240 + 310 + 320 + x. Substituting this expression into the inequality T ≥ 4(280) gives 240 + 310 + 320 + x ≥ 4(280). Therefore, choice D is the correct answer.

Choices A, B, and C are incorrect because they do not correctly capture the relationships between the total number of miles Jaime will ride his bicycle (240 + 310 + 320 + x) and the minimum number of miles he is attempting to bicycle for the four weeks (280 + 280 + 280 + 280).

The two numerical expressions in the given equation can be simplified as √9 = 3 and \(\sqrt{64}\) = 8, so the equation can be rewritten as √x + 3 = 8, or √x = 5. Squaring both sides of the equation gives x = 25.

Choice A is incorrect and may result from a misconception about how to square both sides of √x = 5 to determine the value of x. Choice B is incorrect. The value of √x, not x, is 5. Choice D is incorrect and represents a misconception about the properties of radicals. While it is true that 55 + 9 = 64, it is not true that \(\sqrt{55}\) + √9 = 64.

This question can be answered by making a connection between the table and the algebraic equation. Each row of the table gives a value of x and its corresponding values in both w(x) and t(x). For instance, the first row gives x = 1 and the corresponding values w(1) = −1 and t(1) = −3. The row in the table where x = 2 is the only row that has the property x = w(x) + t(x): 2 = 3 + (−1). Therefore, choice B is the correct answer.

Choice A is incorrect because when x = 1, the equation w(x) + t(x) = x is not true. According to the table, w(1) = −1 and t(1) = −3. Substituting the values of each term when x = 1 gives −1 + (−3) = 1, an equation that is not true. Choice C is incorrect because when x = 3, the equation w(x) + t(x) = x is not true. According to the table, w(3) = 4 and t(3) = 1. Substituting the values of each term when x = 3 gives 4 + 1 = 3, an equation that is not true. Choice D is incorrect because when x = 4, the equation w(x) + t(x) = x is not true. According to the table, w(4) = 3 and t(4) = 3. Substituting the values of each term when x = 4 gives 3 + 3 = 4, an equation that is not true.

By properties of multiplication, the formula n = h \(\iota\) 7 can be rewritten as n = (7h)\(\iota\). To solve for in terms of n and h, divide both sides of the equation by the factor 7h.

Solving this equation for \(\iota\) gives \(\iota=\frac{n}{7h}\).

Choices A, B, and D are incorrect and may result from algebraic errors when rewriting the given equation.

Substituting the given value of y = 18 into the equation x = ²⁄₃ yields ? = (²⁄₃) (18), or x = 12. The value of the expression 2x − 3 when x = 12 is 2(12) − 3 = 21.

Choice B is incorrect. If 2x − 3 = 15, then adding 3 to both sides of the equation and then dividing both sides of the equation by 2 yields x = 9. Substituting 9 for x and 18 for y into the equation x = ²⁄₃ y yields 9 = ²⁄₃18=12, which is false. Therefore, the value of 2x − 3 cannot be 15. Choices C and D are also incorrect. As with choice B, assuming the value of 2x – 3 is 12 or 10 will lead to a false statement.

Choice D is correct. In the xy-plane, the graph of the equation y = mx + b, where m and b are constants, is a line with slope m and y-intercept (0, b). Therefore, the graph of y = 2x − 5 in the xy-plane is a line with slope 2 and a y-intercept (0, −5). Having a slope of 2 means that for each increase in x by 1, the value of y increases by 2. Only the graph in choice D has a slope of 2 and crosses the y-axis at (0, −5). Therefore, the graph shown in choice D must be the correct answer.

Choices A, B, and C are incorrect. The graph of y = 2x − 5 in the xy-plane is a line with slope 2 and a y-intercept at (0, −5). The graph in choice A crosses the y-axis at the point (0, 2.5), not (0,−5), and it has a slope of ¹⁄₂, not 2. The graph in choice B crosses the y-axis at (0, −5); however,the slope of this line is −2, not 2. The graph in choice C has a slope of 2; however, the graph crosses the y-axis at (0, 5), not (0, −5).

Choice A is correct. The right side of the equation can be multiplied using the distributive property: (px + t)(px − t) = p² x²− ptx + ptx − t². Combining like terms gives p²x²− t². Substituting this expression for the right side of the equation gives 4x² − 9 = p² x² − t², where p and t are constants. This equation is true for all values of x only when 4 = p² and 9 = t². If 4 = p², then p = 2 or p = −2. Therefore, of the given answer choices, only 2 could be the value of p.

Choices B, C, and D are incorrect. For the equation to be true for all values of x, the coefficients of x² on both sides of the equation must be equal; that is, 4 = p². Therefore, the value of p cannot be 3, 4, or 9.

For a complex number written in the form a + bi, a is called the real part of the complex number and b is called the imaginary part. The sum of two complex numbers, a + bi and c + di, is found by adding real parts and imaginary parts, respectively; that is, (a + bi) + (c + di) = (a + c) + (b + d)i. Therefore, the sum of 2 + 3i and 4 + 8i is (2 + 4) + (3 + 8)i = 6 + 11i.

Choice A is incorrect and is the result of disregarding i and adding all parts of the two complex numbers together, 2 + 3 + 4 + 8 = 17. Choice B is incorrect and is the result of adding all parts of the two complex numbers together and multiplying the sum by i. Choice D is incorrect and is the result of multiplying the real parts and imaginary parts of the two complex numbers, (2)(4) = 8 and (3)(8) = 24, instead of adding those parts together.

Since Fertilizer A contains 60% filler materials by weight, it follows that x pounds of Fertilizer A consists of 0.6x pounds of filler materials. Similarly, y pounds of Fertilizer B consists of 0.4y pounds of filler materials. When x pounds of Fertilizer A and y pounds of Fertilizer B are combined, the result is 240 pounds of filler materials. Therefore, the total amount, in pounds, of filler materials in a mixture of x pounds of Fertilizer A and y pounds of Fertilizer B can be expressed as 0.6x + 0.4y = 240.

Choice A is incorrect. This choice transposes the percentages of filler zaterials for Fertilizer A and Fertilizer B. Fertilizer A consists of 0.6x pounds of filler materials and Fertilizer B consists of 0.4y pounds of filler materials. Therefore, 0.6x + 0.4y is equal to 240, not 0.4x + 0.6y. Choice C is incorrect. This choice incorrectly represents how to take the percentage of a value mathematically. Fertilizer A consists of 0.6x pounds of filler materials, not 60x pounds of filler materials, and Fertilizer B consists of 0.4y pounds of filler materials, not 40y pounds of filler materials. Choice D is incorrect. This choice transposes the percentages of filler materials for Fertilizer A and Fertilizer B and incorrectly represents how to take the percentage of a value mathematically.

The total amount T, in dollars, Salim will pay for n tickets is given by T = 15n + 12, which consists of both a per-ticket charge and a one-time service fee. Since n represents the number of tickets that Salim purchases, it follows that 15n represents the price, in dollars, of n tickets. Therefore, 15 must represent the per-ticket charge. At the same time, no matter how many tickets Salim purchases, he will be charged the $12 fee only once. Therefore, 12 must represent the amount of the service fee, in dollars.

Choice A is incorrect. Since n represents the total number of tickets that Salim purchases, it follows that 15n represents the price, in dollars, of n tickets, excluding the service fee.Therefore, 15, not 12, must represent the price of 1 ticket. Choice C is incorrect. If Salim purchases only 1 ticket, the total amount, in dollars, Salim will pay can be found by substituting n = 1 into the equation for T. If n = 1, T = 15(1) + 12 = 27. Therefore, the total amount Salim will pay for one ticket is $27, not $12. Choice D is incorrect. The total amount, in dollars, Salim will pay for n tickets is given by 15n + 12. The value 12 represents only a portion of this total amount. Therefore, the value 12 does not represent the total amount, in dollars, for any number of tickets.