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##### 2.1 Solving Linear Equations and Inequalities

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**In your group, write down the things you might need to do or**consider when you’re simplifying algebraic expressions.**Solve.**1. 2(3x – 1) = 34 2. 4y – 9 – 6y = 2(y + 5) – 3**You have solved equations that have a single solution.**Equations may also have infinitely many solutions.An equation that is true for all values of the variable, such as x = x, is an identity. An equation that has no solutions, such as 3 = 5, is a contradiction because there are no values that make it true.**Solve.**1.r + 8 – 5r = 2(4 – 2r) 2. –4(2m + 7) = (6 – 16m)**The graph of an inequality is the solution set, the set of**all points on the number line that satisfy the inequality. • Solve inequalities the same way you do equations, with one important difference. • If you multiply or divide both sides by a negative number, you must reversetheinequality symbol.**Why do you think you have to reverse the inequality sign**when you multiply by a negative number? HINT: What does a negative sign do to a number on the number line? Consider location and position.**These properties also apply to inequalities expressed with**>, ≥, and ≤.**Helpful Hint**• To check an inequality, test • the value being compared with x • a value less than that, and • a value greater than that.**Example 5: Solving Inequalities**–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 Solve and graph: 8a –2 ≥ 13a + 8**–6 –5 –4 –3 –2 –1 0 1 2**3 Solve and graph: x + 8 ≥ 4x + 17**Stacked cups are to be placed in a pantry. One cup is 3.25**in. high and each additional cup raises the stack 0.25 in. How many cups fit between two shelves 14 in. apart?**Bob has 3 times as much money as Amy has, and Sam has $5**more than Bob has. Bob, Amy, and Sam have a total of $75. Write an equation that can be used to find out how much money Amy has? How much money does Sam have?