Section 3.1 Inequalities – Solve and Graph Inequalities Symbols: > Greater Than < Less Than ≥ Greater than or equal to ≤ Less than or equal to

Different Forms to Express the Inequalities Using the inequality symbols Examples: 𝑥<4, −2≤𝑥≤2, 𝑦≥−3 Using the set builder notation {𝑥|𝑥<4}, {𝑥|−2≤𝑥≤2}, {𝑦|𝑦≥−3}

Different Forms to Express the Inequalities Using the number line Using the interval notation

Interval Notation A notation for representing an interval as a pair of numbers. The numbers are the endpoints of the interval. Parentheses are used to show the endpoints are excluded. Brackets are used to show the endpoints are included. For example, [3, 8) is the interval of real numbers between 3 and 8, including 3 and excluding 8.

Interval Notation The left end must be smaller than the right end. Only use parenthesis when the end is ∞ (read "infinity" ) 𝑜𝑟−∞ (read "negative infinity"). −∞, ∞ means all real numbers. ℝ also means all real numbers. (𝑎, 𝑏) is an open interval. [𝑎, 𝑏] is a closed interval.

Finite Inequalities: Both Ends Are Real Numbers If the end point is excluded, use an empty dot. If the end point is included (can be equal to), use a solid dot.

Practice: Finite Intervals Interval Notation Inequality −1<𝑥<4 −2≤𝑥≤2 0<𝑥≤7 2≤𝑥<6 Graph

Infinite Inequalities: At Least One End Goes to Infinity

Practice: Infinite Intervals Interval notation Graph Inequality 𝑥>3 𝑥≥−4 𝑥<1 𝑥≤−2

Practice Which of the following choices corresponds to the graph? B. [3,∞) C. (−∞,3] D. (−∞,3) Which of the following choices corresponds to the graph? (−5,3) [−5,3] (−5,3] [−5,3)

Practice Choose the graph which corresponds to the interval 𝑥>2.

Examples Express the set {𝑥 | −8≤𝑥≤−5 }using interval notation. Solution: [−8, −5] Write the set {𝑥∣0<𝑥≤1} using interval notation Solution: (0, 1]

Solving Inequalities We can add or subtract a number from both sides and keep the inequality relation unchanged Ex. 5>2 5+3>2+3 or 5−3>2−3 We can also multiply or divide both sides by a POSITIVE number and keep the inequality relation unchanged 6>4 6∙3>4∙3 or 6÷2>4÷2

Solving Inequalities −2𝑥<4 𝑥> 4 −2 𝑥>−2 But when we multiply or divide both sides by a NEGATIVE number, WE MUST REVERSE the inequality signs. Ex. 5>2 5∙ −3 <2∙ −3 or 5÷ −1 <2÷ −1 Example: −2𝑥<4 𝑥> 4 −2 𝑥>−2

Example Solve the inequality 16𝑥+8≤−19. Solution: 16𝑥≤−19−8 16𝑥≤−27 𝑥≤− 27 16 Enter your answer as an integer or a reduced fraction in the form A/B.

Example Graph the inequality 4𝑥+8≥20 Solution: 4𝑥≥20−8 𝑥≥ 12 4 𝑥≥3 4𝑥≥20−8 𝑥≥ 12 4 𝑥≥3 You need to solve the inequality and express your solution on the number line. Note: When you divide by positive 4, you don’t change the direction of the inequality sign. The end point 3 is included (can be equal to), so use the solid dot.

Example Solution: −35+5≤6𝑎 −30≤6𝑎 −30 6 ≤𝑎 −5≤𝑎 Solve the following inequality. Write the answer as an inequality, and graph the solution set. −35≤6𝑎−5 Solution: −35+5≤6𝑎 −30≤6𝑎 −30 6 ≤𝑎 −5≤𝑎 Or you can do it this way: −6𝑎≤−5+35 −6𝑎≤30 𝑎≥ 30 −6 𝑎≥−5 -5 Note that you get the same answer either way. But if you use the way on the right, you need to divide by a negative 6 in order to isolate the variable 𝑎.

Example Graph the inequality −8𝑥+6≥2𝑥−2(6𝑥+1) Solution: −8𝑥+6≥2𝑥−12𝑥−2 −8𝑥+6≥−10𝑥−2 −8𝑥+10𝑥≥−2−6 2𝑥≥−8 𝑥≥ −8 2 𝑥≥−4 I’ll leave you to graph this solution on a number line. You need to solve the inequality and express your solution on the number line. Note: When you divide by positive 2, you don’t change the direction of the inequality sign. The end point -4 is included (can be equal to), so use the solid dot.

Example Solve the inequality −2𝑤−7>7(𝑤+5)+3. Solution: −2𝑤−7>7𝑤+35+3 −2𝑤−7𝑤>38+7 −9𝑤>45 𝑤< 45 −9 𝑤<−5 Note: When you divide by negative 9, you need to change the direction of the inequality sign.

Compound Inequality

Compound Inequality Example: Solve the inequality. Then graph the solution and give interval notation. 1≤−4𝑥+5<13 Solution: 1−5≤−4𝑥+5−5<13−5 −4≤−4𝑥<8 −4 −4 ≥𝑥> 8 −4 1≥𝑥>−2 In interval notation: (−2, 1]. -2 1 We want to isolate x in the middle. In this problem: Step 1: we need to subtract 5 on three places Step2: we need to divide by -4 on three places. Since we divide by a negative number, we need to inverse the inequality signs.

Example---Compound Inequality Solve and Graph: 13𝑥+3<7𝑥+21 AND −2+9𝑥≤11𝑥+4 13𝑥+3<7𝑥+21 13𝑥−7𝑥<21−3 6𝑥<18 𝑥< 18 6 𝑥<3 −2+9𝑥≤11𝑥+4 9𝑥−11𝑥≤4+2 −2𝑥≤6 𝑥≥ 6 −2 𝑥≥−3 So the solution must be: 𝑥<3 and 𝑥≥−3. It is the intersection of real numbers that are greater than or equal to -3 and less than 3. -3 3