Graph each inequality. Write an inequality for each situation. 1. The temperature must be at least –10°F. 2. The temperature must be no more than 90°F. x ≥ –10 x ≤ 90 Solve each equation. 3. x – 4 = = x – – Warm Up 5. –10 6.2x – 5 = –17x =–6 7.x = y – 21 = 4 – 2y y = (3z + 1) = – 2(z + 3)z = –1

S OLVING I NEQUALITIES One-Step; Multi-Step; Variables on Both Sides

Solve one-step inequalities by using addition. Solve one-step inequalities by using subtraction. Solve one-step inequalities by using multiplication. Solve one-step inequalities by using division. Solve inequalities that contain more than one operation. Solve inequalities that contain variable terms on both sides. Objectives

Solving one-step inequalities is much like solving one-step equations. To solve an inequality, you need to isolate the variable using the properties of inequality and inverse operations. Helpful Hint Use an inverse operation to “undo” the operation in an inequality. If the inequality contains addition, use subtraction to undo the addition. Remember, solving inequalities is similar to solving equations. To solve an inequality that contains multiplication or division, undo the operation by dividing or multiplying both sides of the inequality by the same number.

d – 5 > –7 Since 5 is subtracted from d, add 5 to both sides to undo the subtraction. Draw an empty circle at –2. Shade all numbers greater than –2 and draw an arrow pointing to the right. +5 d + 0 > –2 d > –2 d – 5 > –7 Using Addition and Subtraction to Solve Inequalities Solve the inequality and graph the solutions. –10 –8 –6–4 –

Multiplying or Dividing by a Positive Number Solve the inequality and graph the solutions. 7x > –42 > 1x > –6 Since x is multiplied by 7, divide both sides by 7 to undo the multiplication. x > –6 –10 –8 –6–4 –

If you multiply or divide both sides of an inequality by a negative number, the resulting inequality is not a true statement. You need to reverse the inequality symbol to make the statement true. Inequalities that contain more than one operation require more than one step to solve. Use inverse operations to undo the operations in the inequality one at a time. To solve more complicated inequalities, you may first need to simplify the expressions on one or both sides by using the order of operations, combining like terms, or using the Distributive Property.

Multiplying or Dividing by a Negative Number Solve the inequality and graph the solutions. –12x > 84 x < –7 Since x is multiplied by –12, divide both sides by –12. Change > to <. –10 –8 –6–4 – –12–14 –7

Since x is divided by –3, multiply both sides by –3. Change to Multiplying or Dividing by a Negative Number Solve the inequality and graph the solutions. 24  x(or x  24)

Caution! Do not change the direction of the inequality symbol just because you see a negative sign. For example, you do not change the symbol when solving 4x < –24.

Solving Multi-Step Inequalities Solve the inequality and graph the solutions b > 61 –45 2b > 16 b > Since 45 is added to 2b, subtract 45 from both sides to undo the addition. Since b is multiplied by 2, divide both sides by 2 to undo the multiplication.

Solve the inequality and graph the solutions (x + 4) > x + 8 > 3 2x + 11 > 3 – 11 2x > –8 x > –4 Distribute 2 on the left side. Combine like terms. Since 11 is added to 2x, subtract 11 from both sides to undo the addition. Since x is multiplied by 2, divide both sides by 2 to undo the multiplication. – 10 –8–8 –6–4 – Simplifying Before Solving Inequalities

Solve each inequality and graph the solutions < t + 9 t > –4 a ≤ –8 Warm Up 1. 2x = 7x + 15x = –3 2. 3(p – 1) = 3p + 2 no solution

Pg 20: #1, 2 Pg 21: # 1-4 Pg 22: #1, 3, 5, 7

Solve the inequality and graph the solutions. 5t + 1 < –2t – 6 +2t 7t + 1 < –6 – 1 < –1 7t < –7 7 t < –1 –5 –4–4 –3–2 – To collect the variable terms on one side, add 2t to both sides. Since 1 is added to 7t, subtract 1 from both sides to undo the addition. Since t is multiplied by 7, divide both sides by 7 to undo the multiplication. Solving Inequalities with Variables on Both Sides

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. –12–9–6–303

There are special cases of inequalities called identities and contradictions.

Identities and Contradictions Solve the inequality. 2x – 7 ≤ 5 + 2x –2x –7 ≤ 5 Subtract 2x from both sides. True statement. The inequality 2x − 7 ≤ 5 + 2x is an identity. All values of x make the inequality true. Therefore, all real numbers are solutions.

2(3y – 2) – 4 ≥ 3(2y + 7) 2(3y) – 2(2) – 4 ≥ 3(2y) + 3(7) 6y – 4 – 4 ≥ 6y y – 8 ≥ 6y + 21 Distribute 2 on the left side and 3 on the right side. Identities and Contradictions Solve the inequality. –6y –8 ≥ 21 Subtract 6y from both sides. False statement.  No values of y make the inequality true. There are no solutions.

Lesson Quiz: Part I Solve each inequality and graph the solutions – 2x ≥ 21x ≤ –4 2. – < 3p p > – < – 2(3 – t) t > 7 4.

Lesson Quiz: Part II 5. A video store has two movie rental plans. Plan A includes a $25 membership fee plus $1.25 for each movie rental. Plan B costs $40 for unlimited movie rentals. For what number of movie rentals is plan B less than plan A? more than 12 movies

Lesson Quiz: Part III Solve each inequality and graph the solutions. 1. t < 5t + 24t > –6 2. 5x – 9 ≤ 4.1x – 81x ≤ –80 b < b + 4(1 – b) > b – 9

Lesson Quiz: Part IV 4. Rick bought a photo printer and supplies for $186.90, which will allow him to print photos for $0.29 each. A photo store charges $0.55 to print each photo. How many photos must Rick print before his total cost is less than getting prints made at the photo store? Rick must print more than 718 photos.