Linear Equations and Inequalitites

1. Linear Equations and Inequalitites

2. Algebra/Standard • GLE 0506.3.4 Solve single-step linear equations and inequalities. • Objectives: Given a set or values, identify those that make an inequality a true statement.

3. What’s in the Bag? • “I have 4 pencils in this hand and a number from 0 to 9 pencils in this bag. How could we write an equation representing the total number of pencils I have?” (4 + = )

4. What’s in the Bag? • Distribute What’s in the Bag. • Complete Part I as a whole group. The total number of pencils is a one-digit number. Ask: What are the possible values of ? (1, 2, 3, 4 or 5)The total number of pencils is 9 and ask how many pencils must be in the bag. (5) • Pull out another bag. Explain that 9 pencils were in the bag but that you later discovered a hole in the bag. Some pencils may have fallen out. Write an equation representing this scenario. (9 - Δ = ∩) Have students work with a partner to complete Part 2. • Have students complete Part 3 of What’s in the Bag with a partner. Students should be encouraged to draw a table of values if necessary. (a. x = 8, b. x = 8, c. x = 2)

5. PASS IT ON Divide class into groups of 5 Each member get a card with an algebraic expression . Each member get a card with an algebraic expression Each group lines up with “b=“ card member the first in the line and the “f=“ card member is at the end of the line. Announce a value for “a”. The first member of each group solves the equation by substituting the value for “a” and then passes the answer to the next member . This continues until the last member arrives at the answer and comes to the front of the line.

6. Writing Algebraic Equations Equations are like scales. Both sides of the equation must be equal. We represent this as: ×= 7 × We can represent this as: 2 + × = 5 × Here × = 3.

7. What is ×? 4 3 5 ×

8. Solving Equations Use cups and counters to model and solve algebraic equations • The cup represents an unknown value, x. Place a cup and 5 positive counters on the left Place 9 positive counters on the right side A model of the equation x + 5 = 9. W To solve an equation, they must isolate the variable on one side of the equation.

10. Solve x + 3 = 5 a. Use a cup to represent the unknown value, x. Place 1 cup and 3 yellow counters on the left side of an equation mat. Place 5 yellow counters on the right side of the mat. b. The goal is to determine what’s in the cup. To do so, get the cup by itself on one side of the mat. Then the counters on the other side will be the value of the cup, or x. c. To find the value of x, remove 3 counters from the left side and 3 counters from the right side. Remind the students that they need to remove the same amount from both sides to maintain the balance. d. The cup is by itself on one side of the mat. There are two counters on the other side. Therefore, the solution is 2.

11. 3c = 9 Step 1: Have students place 3 cups on the left side of the mat to represent 3c and then place 9 yellow counters on the right side of the scale to represent 9. Explain that the balance scale now represents 3c = 9, because the left side (3c) and the right side (9) are balanced, or equal. Step 2: Explain that, to find the value of one cup, students must divide the counters equally among the 3 cups. Have students place the counters into 3 equal groups, one at a time, until all the counters are divided equally among the cups. Count how many counters there are in each group. There are 3 counters in each group, so 1 cup (c) = 3 yellow chips; c = 3. Explain to students that they isolated the variable by dividing the counters into each of the 3 cups to find the value of 1 cup. They used division to “undo” multiplication because division is the inverse operation for multiplication.

12. Linear Equations and Inequalities • A linear equation in one variable has a single unknown quantity called a variable represented by a letter. Eg: ‘x’, where ‘x’ is always to the power of 1. This means there is no ‘x² ’ or ‘x³ ’ in the equation. For example, this linear equation:  x + 1 = 4 means that when we add 1 to the unknown value, ‘x’, the answer is equal to 4.

13. Algorithm for Linear Equations And Inequalities • STEP I :  To solve linear equations, you add, subtract, multiply and divide both sides of the equation by numbers and variables, so that you end up with a single variable on one side and a single number on the other side. • STEP II  : As long as you always do the same thing to both sides of the equation, and do the operations in the correct order, you will get to the solution.

14. Linear Equations and Inequalities • EXAMPLE 1: x + 1 = 4 Step 1: we need to subtract 1 from both sides of the equation in order to isolate 'x' x + 1 - 1 = 4 – 1 Step 2: Now simplifying both sides we have  x + 0  = 3    x = 3

15. Inequalities Solving one-step inequalities is much like solving one-step equations. To solve an inequality, you need to isolate the variable using inverse operations. Graphs of inequalities consist of rays. The endpoint of the ray is open if the inequality is exclusive (< or >) and closed if the symbol is inclusive (≤ or ≥ ). The ray points in the direction of the inequality symbol, provided the variable is on the left side of the inequality.

16. Single step linear Inequalities • In the process of solving an equation, we use mathematical simplifications which are governed by the following rules: • RULE 1  Same number may be added to (or subtracted from) both sides of an inequality without changing the sign of inequality. • RULE 2  Both sides of an equation can be multiplied (or divided) by the same positive real number without changing the sign of inequality. However, the sign of inequality is reversed when both sides of an equation are multiplied or divided by a negative number. • RULE 3  Any term of an equation may be taken to the other side with its sign changed without affecting the sign of inequality.

17. Linear Equations and Inequalities • EXAMPLE   Solve the inequality :                  x – 6 > 14        x – 6+ 6 > 14 + 6                x > 20