Complete the work shown to simplify the expression: $5(-2 x-6)$
$5(-2 x)-5(6)$
To simplify the expression \(5(-2x - 6)\), we use the distributive property of multiplication over subtraction. The distributive property states that \(a(b - c) = ab - ac\). In our case, \(a = 5\), \(b = -2x\), and \(c = 6\). Applying this property, we have:
\[5(-2x - 6) = 5 \times (-2x) - 5 \times 6.\]
Next, we perform the multiplication in each term. First, calculate the product of the coefficients of \(x\): \(5 \times (-2) = -10\). Thus, the first term becomes \(-10x\). Second, calculate the product of the constants: \(5 \times 6 = 30\). Subtracting this result gives:
\[-10x - 30.\]
Comparing this result to the form \(\boxed{\phantom{x}}x - 30\) given in the problem, we can see that the coefficient to be placed in the box is \(-10\).
In real life, the distributive property is used in everyday shopping. For example, if you are buying 5 items that each cost \(\$2\) for tax and \(\$6\) for shipping per item, you can calculate the total cost as \(5 \times (\$2 + \$6)\). You could either sum the costs first, \(5 \times \$8 = \$40\), or distribute the 5, \(5 \times \$2 + 5 \times \$6 = \$10 + \$30 = \$40\). This demonstrates how distributing multipliers helps keep complex calculations organized and simple.