1. Algebra 1 Chapter 3 Section 5

2. 3-5 Solving Inequalities With Variables on Both Sides Some inequalities have variable terms on both sides of the inequality symbol. You can solve these inequalities like you solved equations with variables on both sides. Use the properties of inequality to “collect” all the variable terms on one side and all the constant terms on the other side. Step 2: MOVE the SMALLER variable.

3. Example 1: Solving Inequalities with Variables on Both Sides Solve the inequality and graph the solutions. y ≤ 4y + 18 –y 0 ≤ 3y + 18 –18 – 18 –18 ≤ 3y To collect the variable terms on one side, MOVE the Smaller variable. Since 18 is added to 3y, subtract 18 from both sides to undo the addition. Since y is multiplied by 3, divide both sides by 3 to undo the multiplication.

4. Example 1: Continued Solve the inequality and graph the solutions. y ≤ 4y + 18 –6 ≤ y (or y  –6) –10 –8 –6–4 –2 0246810 The solution set is { y:y ≥ –6 }.

5. 4m – 3 < 2m + 6 To collect the variable terms on one side, subtract 2m from both sides. –2m – 2m 2m – 3 < + 6 Since 3 is subtracted from 2m, add 3 to both sides to undo the subtraction. + 3 2m < 9 Since m is multiplied by 2, divide both sides by 2 to undo the multiplication. Example 2: Solving Inequalities with Variables on Both Sides Solve the inequality and graph the solutions.

6. 4m – 3 < 2m + 6 Example 2 Continued Solve the inequality and graph the solutions. 4 5 6 The solution set is { m:m }.

7. You may need to simplify one or both sides of an inequality before solving it. Look for like terms to combine and places to use the Distributive Property.

8. Example 3: Simplify Each Side Before Solving Solve the inequality and graph the solutions. 2(k – 3) > 6 + 3k – 3 2(k – 3) > 3 + 3k Distribute 2 on the left side of the inequality. 2k + 2(–3) > 3 + 3k 2k – 6 > 3 + 3k –2k – 2k –6 > 3 + k To collect the variable terms, subtract 2k from both sides. –3 –9 > k Since 3 is added to k, subtract 3 from both sides to undo the addition.

9. Example 3 Continued –9 > k –12–9–6–303 Solve the inequality and graph the solutions. The solution set is { k:k < –9 }. 2(k – 3) > 6 + 3k – 3

10. Some inequalities are true no matter what value is substituted for the variable. For these inequalities, the solution set is all real numbers. Some inequalities are false no matter what value is substituted for the variable. These inequalities have no solutions. Their solution set is the empty set, ø. If both sides of an inequality are fully simplified and the same variable term appears on both sides, then the inequality has all real numbers as solutions or it has no solutions. Look at the other terms in the inequality to decide which is the case.

11. Example 4: All Real Numbers as Solutions or No Solutions Solve the inequality. 2x – 7 ≤ 5 + 2x The same variable term (2x) appears on both sides. Look at the other terms. For any number 2x, subtracting 7 will always result in a lower number than adding 5. All values of x make the inequality true. All real numbers are solutions.

12. 2(3y – 2) – 4 ≥ 3(2y + 7) Solve the inequality. Example 4: All Real Numbers as Solutions or No Solutions Distribute 2 on the left side and 3 on the right side and combine like terms. 6y – 8 ≥ 6y + 21 The same variable term (6y) appears on both sides. Look at the other terms. For any number 6y, subtracting 8 will never result in a higher number than adding 21. No values of y make the inequality true. There are no solutions. The solution set is .