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Solve the equations below for a function $h$ where $h(x)=-4x+7\text{.}$ Check each answer and state the solution set.

Solution

To solve for $x\text{,}$ we will substitute $h(x)$ with its formula, $-4x+7\text{:}$
$\begin{aligned} h(x) \amp= -1\\ \substitute{-4x+7} \amp= -1\\ -4x \amp= -8\\ x \amp=2 \end{aligned}$
To check, verify that $h(2)=-1\text{:}$
$\begin{aligned} h(2) \amp= -4(2)+7\\ \amp= -8+7\\ \amp\stackrel{\checkmark}{=} -1 \end{aligned}$
The solution is $2\text{,}$ and the solution set is $\{2\}\text{.}$

To solve for $x\text{,}$ we will substitute $h(x)$ with its formula:
$\begin{aligned} h(x) \amp= 0\\ \substitute{-4x+7} \amp= 0\\ -4x \amp= -7\\ x \amp=\frac{7}{4} \end{aligned}$
To check, we will verify that $h\mathopen{}\left(\frac{7}{4}\right)\mathclose{}=0\text{:}$
$\begin{aligned} h\mathopen{}\left(\frac{7}{4}\right)\mathclose{} \amp = -4\left(\frac{7}{4}\right)+7\\ \amp= -7+7\\ \amp\stackrel{\checkmark}{=} 0 \end{aligned}$
The solution is $\frac{7}{4}\text{,}$ and the solution set is $\left\{\frac{7}{4}\right\}\text{.}$

To solve for $x\text{,}$ we will substitute $h(x)$ with its formula:
$\begin{aligned} h(x) \amp= 11\\ \substitute{-4x+7} \amp= 11\\ -4x \amp= 4\\ x \amp=-1 \end{aligned}$
To check, we will verify that $h(-1)=11\text{:}$
$\begin{aligned} h(-1) \amp= -4(-1)+7\\ \amp= 4+7\\ \amp\stackrel{\checkmark}{=} 11 \end{aligned}$
The solution is $-1\text{,}$ and the solution set is $\{-1\}\text{.}$