Question

question
complete the sentence based on the equation shown below.
$h(-3x - 9) = -9(x + 3)$
answer attempt 1 out of 2
this equation will have infinitely many solutions when $h = \square$ because
you get infinitely many solutions when you have $\quad$ number of
xs on either side of the equation and
$\quad$

Explanation:

Step1: Analyze the equation structure

We have the equation \( h(-3x - 9)=-9(x + 3) \). First, factor out \(-3\) from the left - hand side's argument: \(-3x-9=-3(x + 3)\). So the equation becomes \( h(-3(x + 3))=-9(x + 3) \).

Step2: Let \( u=-3(x + 3) \), then the equation is \( h(u)=-9\times\frac{u}{-3}\) (since \( u=-3(x + 3)\), then \( x + 3=-\frac{u}{3}\))

Simplify the right - hand side: \(-9\times\frac{u}{-3}=3u\). So we want \( h(u) = 3u \). Substituting back \( u=-3x - 9 \), we can see that for the equation \( h(-3x - 9)=-9(x + 3) \) to hold for all \( x \) (infinitely many solutions), we need to find \( h \) such that when we input \(-3x - 9\) into \( h \), we get \(-9(x + 3)\).

Another way: Expand both sides. The left - hand side: Let's assume \( h \) is a linear function of the form \( h(a)=k\times a\) (since the argument of \( h \) and the right - hand side are linear in \( x \)). The argument of \( h \) is \(-3x-9=-3(x + 3)\), and the right - hand side is \(-9(x + 3)\). So if we let \( a=-3(x + 3) \), then we have \( h(a)=\frac{-9(x + 3)}{1}\), and since \( a=-3(x + 3) \), then \( x + 3=-\frac{a}{3}\). Substitute \( x + 3=-\frac{a}{3}\) into the right - hand side: \( h(a)=-9\times(-\frac{a}{3}) = 3a\). So when \( h = 3 \), the left - hand side is \( 3(-3x-9)=-9x - 27\), and the right - hand side is \(-9(x + 3)=-9x-27\). The equation \( 3(-3x - 9)=-9(x + 3) \) simplifies to \(-9x-27=-9x - 27\), which is an identity (true for all \( x \)), so it has infinitely many solutions.

Answer:

The value of \( h \) is \( 3 \). The equation will have infinitely many solutions when \( h = 3 \) because when \( h = 3 \), the left - hand side \( h(-3x - 9)=3(-3x - 9)=-9x-27 \) and the right - hand side \( -9(x + 3)=-9x - 27 \), and the equation \(-9x-27=-9x - 27\) is an identity (it holds for all values of \( x \)), which means there are infinitely many solutions. So \( h = 3 \).