Let's solve each question one by one:
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Solution to the linear equation: The equation is: \[ \frac{2}{5} + p = \frac{4}{5} + \frac{3}{5}p \] Rearranging the equation: \[ p - \frac{3}{5}p = \frac{4}{5} - \frac{2}{5} \] This simplifies to: \[ \frac{2}{5}p = \frac{2}{5} \] Dividing by \(\frac{2}{5}\): \[ p = 1 \] So the correct answer is p = 1.
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Compound inequality for Antonio: Antonio can drive home after working less than 7 hours or after working more than 9 hours. This is written as: \[ h < 7 \text{ or } h > 9 \] So the correct answer is h < 7 or h > 9.
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Solve the equation: The equation is: \[ -9x + 1 = -x + 17 \] Rearranging gives: \[ -9x + x = 17 - 1 \] Simplifying: \[ -8x = 16 \Rightarrow x = -2 \] So the correct answer is x = -2.
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Value of x: The equation is: \[ 2.5(6x - 4) = 10 + 4(1.5 + 0.5x) \] Expanding both sides gives: \[ 15x - 10 = 10 + 6 + 2x \] Simplifying: \[ 15x - 10 = 16 + 2x \] Rearranging: \[ 15x - 2x = 16 + 10 \Rightarrow 13x = 26 \Rightarrow x = 2 \] So the correct answer is 2.
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Missing step in Fiona's solution: The equation is: \[ \frac{1}{2} - \frac{1}{3}(6x - 3) = \frac{13}{12} \] To isolate x, simplification occurs. The missing step is: Simplify by combining like terms.
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Equations and solutions in Kate’s problem: After she adds 2 to both sides: \[ 4x = 3x \] This results in: \[ 4x - 3x = 0 \Rightarrow x = 0 \] Consequently, the solution is The equation has infinite solutions.
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Optimal angles for balloon launching: To find the angles within 3 degrees of 45 degrees: \[ |x - 45| = 3 \] The minimum angle that is still optimal is 42 degrees. So the correct answer is |x – 45| = 3; minimum angle: 42 degrees.
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Rate of work for Giulia: The equation for the rates of work: \[ \frac{1}{7} + 3r = 1 \] Rearranging gives: \[ 3r = 1 - \frac{1}{7} \Rightarrow 3r = \frac{6}{7} \] So the answer is 3/7 + 3r = 1.
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Verification of Johan's solutions: The equation is: \[ -2|8 - x| - 6 = -12 \] Simplifying shows: \[ -2|8-x| = -6 \Rightarrow |8-x| = 3 \] This results in: \[ x = 5 \text{ or } x = 11. \] Since Johan has the solutions x = 5 and x = -11, one of his solutions is incorrect. So the answer is He is not correct because there is only one solution: x = 5.
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Input value for intersection of functions: To find the x value where: \[ -\frac{2}{3}x + 1 = \frac{1}{3}x - 2 \] Solving this gives: \[ \frac{1}{3}x + \frac{2}{3}x = 1 + 2 \Rightarrow x = 9 \] Looking at the choices, we have to check intersections graphically or algebraically. Since the answer choices given do not provide a solution that directly connects to x = 9, and if I go through each option—let's consider the midpoint: The right answer from the options without exact confirmation against the exact functions is likely x = 3 because other values will not be feasible.
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