1. Section 2.5 Solving Linear Inequalities
Chapter 2
Section 2.5
Solving Linear Inequalities

2. Interval Notation
Another way to express and inequality or a number line is to use interval notation. We use a "(" to stand for an open circle and a "[" to stand for a closed circle. We use the symbol  to stand for an arrow.
Algebra (Sets)
Interval Notation
Number Line
𝑥|𝑥>−2
−2,∞
𝑥|𝑥≤6
−∞,6
𝑥|−4≤𝑥<5
−4,5
𝑥|−5<𝑥<1
−5,1
𝑥<0 𝑜𝑟 𝑥≥3
−∞,0 ∪ 3,∞
Intervals that are described in pieces have the symbol ∪ between them and the word "or" between the inequalities.

3. Solve,
2𝑥+3≥11
2𝑥≥11+ −3
2𝑥≥8
𝑥≥ 1 2 ∙8
𝑥≥4
Solve,
7− 1 5 𝑥>8
− 1 5 𝑥>8+ −7
− 1 5 𝑥>1
𝑥<−5∙ 1
𝑥<−5
Solving Single Inequalities
To solve an inequality you isolate the variable just like and equation with one additional rule. The rule is if you need to multiply or divide by a negative number you need to reverse the direction the inequality points. The answer can be given using either algebra, a number line, or interval notation.
Number Line
Number Line
Interval Notation
Interval Notation
4,∞
−∞,−5

4. Solving Double Inequalities
A double inequality (or multiple inequality) has two inequality signs in it. They are solved by isolating the variable in the middle.
Solve,
34≤ −1 4 𝑥+36≤37
34+ −36 ≤ −1 4 𝑥≤37+ −36
−2≤ −1 4 𝑥≤1
−4 ∙ −2 ≥𝑥≥ −4 ∙1
8≥𝑥≥−4
−4≤𝑥≤8
Solve,
−5≤3𝑥−11<7
−5+11≤3𝑥<7+11
6≤3𝑥<18
1 3 ∙6≤𝑥< 1 3 ∙18
2≤𝑥<6
Interval Notation:
−4,8
Interval Notation:
2,6
Number Line
Number Line

5. Solve,
𝑡 3 −𝑡+7≤− 4𝑡 3 +8
3 𝑡 3 −𝑡+7≤− 4𝑡 3 +8
3∙ 𝑡 3 −3𝑡+3∙7≤3 − 4𝑡 3 +3∙8
𝑡−3𝑡+21≤−4𝑡+24
−2𝑡+21≤−4𝑡+24
−2𝑡+21+ −21 ≤−4𝑡+24+ −21
−2𝑡≤−4𝑡+3
−2𝑡+4𝑡≤−4𝑡+3+4𝑡
2𝑡≤3
𝑡≤ 1 2 ∙3
𝑡≤ 3 2
Interval Notation
−∞, 3 2
Number Line