Answer:
for any value of h less than 2, the equation 6x + 18 = h(3x + 9) will have a unique solution for x.
Explanation:
To find the value of the constant h in the equation 6x + 18 = h(3x + 9), we can use the distributive property to simplify the equation.
First, distribute the h to both terms inside the parentheses:
6x + 18 = 3hx + 9h
Next, rearrange the equation to isolate the terms with x on one side and the constant terms on the other side:
6x - 3hx = 9h - 18
Factor out x from the left side of the equation:
x(6 - 3h) = 9h - 18
Now, divide both sides of the equation by (6 - 3h) to solve for x:
x = (9h - 18) / (6 - 3h)
The value of h will determine whether this equation has a unique solution for x. If (6 - 3h) equals zero, the denominator will be zero, and the equation will not have a solution. So we need to find the value of h that makes (6 - 3h) not equal to zero.
To find this value, set (6 - 3h) not equal to zero and solve for h:
6 - 3h ≠ 0
Subtract 6 from both sides:
-3h ≠ -6
Divide both sides by -3 (note that dividing by a negative number reverses the inequality sign):
h < 2
Therefore, for any value of h less than 2, the equation 6x + 18 = h(3x + 9) will have a unique solution for x.