2.1 Solving Linear Equations and

Inequalities

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 reverse the inequality 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 ≤.

To check an inequality, test• the value being compared with x • a value less than that, and• a value greater than that.

Helpful Hint

Solve and graph:

8a –2 ≥ 13a + 8

Example 5: Solving Inequalities

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

Solve and graph:

x + 8 ≥ 4x + 17

–6 –5 –4 –3 –2 –1 0 1 2 3

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?