## College Algebra (11th Edition)

$-2$
$\bf{\text{Solution Outline:}}$ To evaluate the given summation expression, $\displaystyle\sum_{i=1}^4 (-3x_i-2) ,$ substitute $i$ with the values from $1$ to $4 .$ Then substitute the given values for each $x_i$'s $\bf{\text{Solution Details:}}$ Substituting $i$ with the numbers from $1$ to $5 ,$ the given expression is equivalent to \begin{array}{l}\require{cancel} (-3x_1-2)+(-3x_2-2)+(-3x_3-2)+(-3x_4-2) \\\\= -3x_1-2-3x_2-2-3x_3-2-3x_4-2 \\\\= (-3x_1-3x_2-3x_3-3x_4)+(-2-2-2-2) \\\\= (-3x_1-3x_2-3x_3-3x_4)+(-8) \\\\= (-3x_1-3x_2-3x_3-3x_4)-8 .\end{array} Factoring the common factor, $-3,$ in the expression $(-3x_1-3x_2-3x_3-3x_4-3x_5),$ results to \begin{array}{l}\require{cancel} -3(x_1+x_2+x_3+x_4)-8 .\end{array} Using the given values $x_1=-2,x_2=-1,x_3=0,$ and $x_4=1,$ the expression above evaluates to \begin{array}{l}\require{cancel} -3(-2+(-1)+0+1)-8 \\\\= -3(-2-1+0+1)-8 \\\\= -3(-2)-8 \\\\= 6-8 \\\\= -2 .\end{array}