Solve the inequality -7 < 5x - 2x - 3. Represent the solution on a real number line. If g(x) = x2 + 2 and h(x) = x + 2, find g(2) and h^-1(x).

Hey Miriam, good to see you again. Here are the solutions to the problems: Problem 1: a) Solve the inequality -7 < 5x - 2x - 3. Step 1: Combine like terms on the right side of the inequality. -7 < (5x - 2x) - 3 -7 < 3x - 3 Step 2: Add 3 to both sides of the inequality to isolate the term with x. -7 + 3 < 3x - 3 + 3 -4 < 3x Step 3: Divide both sides by 3 to solve for x. (-4)/(3) < (3x)/(3) -(4)/(3) < x The solution is x > -(4)/(3). x > -(4)/(3) b) Represent the solution of (a) on a real number line. To represent x > -(4)/(3) on a real number line, draw a number line. Place an open circle at the point -(4)/(3) (which is approximately -1.33). Draw an arrow extending to the right from this open circle, indicating that all numbers greater than -(4)/(3) are part of the solution. Problem 2: The functions g and h are defined on R as follows: g(x) = x^2 + 2 h(x) = x + 2 a) Find g(2). Step 1: Substitute x=2 into the function g(x). g(2) = (2)^2 + 2 g(2) = 4 + 2 g(2) = 6 g(2) = 6 b) Find h^-1(x). Step 1: Let y = h(x). y = x + 2 Step 2: Swap x and y to find the inverse relation. x = y + 2 Step 3: Solve for y to express the inverse function. y = x - 2 So, the inverse function is h^-1(x) = x - 2. h^-1(x) = x - 2 c) Find hg(x). (This means h(g(x))) Step 1: Substitute the expression for g(x) into h(x). h(g(x)) = h(x^2 + 2) Step 2: Replace x in the function h(x) with the expression (x^2 + 2). h(x^2 + 2) = (x^2 + 2) + 2 h(x^2 + 2) = x^2 + 4 h(g(x)) = x^2 + 4 Problem 3: Given y varies directly as x and that y = 10 when x = (1)/(2). a) Find the relation between x and y. Step 1: Write the general equation for direct variation. y = kx where k is the constant of proportionality. Step 2: Substitute the given values y=10 and x=(1)/(2) into the equation to find k. 10 = k ((1)/(2)) Step 3: Solve for k. 10 × 2 = k k = 20 Step 4: Write the relation between x and y using the value of k. y = 20x y = 20x b) Hence find the value of x when y = 6. Step 1: Use the relation found in part (a). y = 20x Step 2: Substitute y=6 into the relation. 6 = 20x Step 3: Solve for x. x = (6)/(20) x = (3)/(10) x = (3)/(10) What's next? Send 'em! 📸