To solve the inequality 4x - 3 < 2x + 5, we will isolate the variable x: Step 1: Subtract 2x from both sides of the inequality 2x - 3 < 5 Step 2: Add 3 to both sides of the inequality 2x < 8 Step 3: Divide by 2 x < 4 Thus, the inequality 4x - 3 < 2x + 5 is true for all values of x less than 4, which corresponds to option A.

To solve this system of equations, we can use substitution or elimination. We will use substitution in this case:

From the first equation, y = 6 - x.

Now, substitute this expression for y in the second equation:

x² + (6 - x)² = 36

Expand and simplify:

x² + 36 - 12x + x² = 36

Combine like terms:

2x² - 12x = 0

Factor out 2x:

2x(x - 6) = 0

Set each term equal to 0:

2x = 0 => x = 0
x - 6 = 0 => x = 6

Now substitute x back into the first equation to find the corresponding values for y:

x = 0: y = 6 - 0 = 6 (0, 6)
x = 6: y = 6 - 6 = 0 (6, 0)

Thus, the solutions are (0, 6) and (6, 0), which corresponds to option D.

Let x and y be the two numbers.

Write the system of linear equations using the given information:
x + y = 24
x - y = 10

Add the two equations together to eliminate y:
2x = 34

Divide by 2 to solve for x:
x = 17

Now substitute x into the first equation:
17 + y = 24

Subtract 17 from both sides of the equation to solve for y:
y = 7

Thus, the two numbers are 17 and 7, which corresponds to option C.

To solve the absolute value equation |3x - 4| = 8, we need to split the equation into two separate equations:

3x - 4 = 8 and 3x - 4 = -8

For the first equation, add 4 to both sides:
3x = 12
Divide by 3:
x = 4

For the second equation, add 4 to both sides:
3x = -4
Divide by 3:
x = -4/3

Thus, the solution to the absolute value equation |3x - 4| = 8 is x = 4 and x = -4/3, which corresponds to option D.

To find the vertex of the quadratic function f(x) = -2x² + 8x + 1, we can use the following formula for the x-coordinate of the vertex:

x = -b / 2a

In this function, a = -2 and b = 8, so:

x = -8 / (2 * -2) = -8 / -4 = 2

Now, plug in this x-value into the function to find the corresponding y-value:

f(2) = -2(2)² + 8(2) + 1 = -8 + 16 + 1 = 9

Thus, the vertex of the given quadratic function is (2, 9), which corresponds to option B.