170 CHAPTER 1 Equations and Inequalities 3 0 5 – Figure 10 EXAMPLE 2 Solving a Linear Inequality Solve 4 - 3x … 7 + 2x. Give the solution set in interval notation. SOLUTION 4 - 3x … 7 + 2x 4 - 3x - 4 … 7 + 2x - 4 Subtract 4. -3x … 3 + 2x Combine like terms. -3x - 2x … 3 + 2x - 2x Subtract 2x. -5x … 3 Combine like terms. -5x -5 Ú 3 -5 Divide by -5. Reverse the direction of the inequality symbol. x Ú - 3 5 a -b = - a b See Figure 10 for the graph. In interval notation, the solution set is C - 3 5 , ∞B. S Now Try Exercise 15. Three-Part Inequalities The inequality -2 65 + 3x 620 says that 5 + 3x is between -2 and 20. This inequality is solved using an extension of the properties of inequality given earlier, working with all three expressions at the same time. At least equal translates as Ú. EXAMPLE 3 Finding the Break-Even Point If the revenue and cost of a certain product are given by R = 4x and C = 2x + 1000, where x is the number of units produced and sold, at what production level does R at least equal C? SOLUTION Set R Ú C and solve for x. R Ú C 4x Ú 2x + 1000 Substitute. 2x Ú 1000 Subtract 2x. x Ú 500 Divide by 2. The break-even point is at x = 500. This product will at least break even if the number of units produced and sold is in the interval 3500, ∞2. S Now Try Exercise 25. A product will break even, or begin to produce a profit, only if the revenue from selling the product at least equals the cost of producing it. If R represents revenue and C is cost, then the break-even point is the point where R = C.
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