2.1 Solving Equations and Inequalities. The local phone company charges $12.95 a month for the first 200 of air time, plus $0.07 for each additional minute.

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Presentation transcript:

2.1 Solving Equations and Inequalities

The local phone company charges $12.95 a month for the first 200 of air time, plus $0.07 for each additional minute. If Nina’s bill for the month was $14.56, how many additional minutes did she use?

monthly charge plus additional minute charge times number of additional minutes total charge + = Let m represent the number of additional minutes that Nina used. m * = Model

Solve m = 14.56

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?

c = cups fit between the 14 in. shelves. Solve.

Solve 4(m + 12) = –36

Solve –3(5 – 4r) = –9.

If there are variables on both sides of the equation, (1) simplify each side. (2) collect all variable terms on one side and all constants terms on the other side. (3) isolate the variables as you did in the previous problems.

Solve 3k– 14k + 25 = 2 – 6k – 12.

These equations have a single solution. However, equations may also have infinitely many solutions or no solution. 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 3v – 9 – 4v = –(5 + v). The equation has no solution. The solution set is the empty set, which is represented by the symbol.

Solve 2(x – 6) = –5x – x. The solutions set is all real number, or .

Solve 3(2 –3x) = –7x – 2(x –3). The solutions set is all real numbers, or .

An inequality is a statement that compares two expressions by using the symbols, ≤, ≥, or ≠. The graph of an inequality is the solution set, the set of all points on the number line that satisfy the inequality. The properties of equality are true for inequalities, with one important difference. If you multiply or divide both sides by a negative number, you must reverse the inequality symbol.

To check an inequality, test the value being compared with x a value less than that, and a value greater than that. mean an open circle ≤ or ≥ mean a closed circle If it’s EATING the variable, shade to the right. If it’s EATING the number, shade to the left. Helpful Hints

Solve and graph 8a –2 ≥ 13a + 8.

Solve 5(x – 6) = 3x – x.

Solve 3(2 –3p) = 42.

Solve 3(w + 7) – 5w = w + 12.

Solve and graph x + 8 ≥ 4x + 17.