2.3: Inequalities and Compound Statements | Algebra 1 Honors

How it works

An inequality describes a set of allowed values. Adding or subtracting the same quantity preserves its direction. Multiplying or dividing by a negative number reverses the direction. An AND statement requires both conditions; an OR statement permits either one.

a<b,\ c<0\quad\Longrightarrow\quad ac>bc

Worked examples

Example 1

Solve $\displaystyle -3x+4<13$ .

Subtract $\displaystyle 4$ : $\displaystyle -3x<9$ . Dividing by $\displaystyle -3$ reverses the sign: $\displaystyle x>-3$ .

Example 2

Solve $\displaystyle -2\le3x+1<10$ .

Subtract $\displaystyle 1$ throughout: $\displaystyle -3\le3x<9$ . Divide by $\displaystyle 3$ : $\displaystyle -1\le x<3$ .

Problems

Problem 2.3.1

Solve $\displaystyle 5-2x\ge11$ .

Show worked solution

$\displaystyle -2x\ge6$ . Divide by $\displaystyle -2$ and reverse the relation: $\displaystyle x\le-3$ .

Problem 2.3.2

Solve $\displaystyle 2<x+1\le6$ .

Show worked solution

Subtract $\displaystyle 1$ throughout: $\displaystyle 1<x\le5$ . The left endpoint is excluded and the right is included.

Problem 2.3.3

The diagram shows an interval. Write it as an inequality and in interval notation.

A blue interval runs from an open endpoint at negative 1 to a filled endpoint at 3. — Endpoint fill indicates whether equality is included.

Show worked solution

The open point at $\displaystyle -1$ is excluded and the filled point at $\displaystyle 3$ is included. The set is $\displaystyle -1<x\le3$ , or $\displaystyle (-1,3]$ .

A blue interval runs from an open endpoint at negative 1 to a filled endpoint at 3. The interval is (−1, 3]. — The interval is (−1, 3].

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 2.3.1

Problem 2.3.2

Problem 2.3.3

Need help with this topic?