Question

1. 478 > -5 + 7(8v + 5)

number line with ticks at 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, arrows left and right

1. 5cut off, with a number line starting at -1, arrow left

Explanation:

Step1: Expand the right - hand side

First, we use the distributive property \(a(b + c)=ab+ac\) to expand \(7(8v + 5)\). So \(7(8v+5)=7\times8v+7\times5 = 56v+35\). Then the inequality becomes \(478>- 5 + 56v+35\).

Step2: Simplify the right - hand side

Combine like terms on the right - hand side. \(-5 + 35=30\), so the inequality is \(478>56v + 30\).

Step3: Subtract 30 from both sides

Subtract 30 from both sides of the inequality: \(478-30>56v+30 - 30\). Calculating the left - hand side, \(478 - 30 = 448\), so we get \(448>56v\).

Step4: Divide both sides by 56

Divide both sides of the inequality \(448>56v\) by 56. \(\frac{448}{56}>v\), and \(\frac{448}{56} = 8\). So \(v < 8\).

Answer:

The solution to the inequality \(478>-5 + 7(8v + 5)\) is \(v < 8\). On the number line, we would draw an open circle at \(v = 8\) and shade the region to the left of 8.