P-3 Linear Equations and Inequalities

Vocabulary Linear Equation in one variable. Ax + B = C A ≠ 0 B and C are constants You’ve seen this before! 4x – 2 = 6 Give me another example of a linear equation in one variable x = 8 ÷ 4 ÷4 x = 2 x = 2

Linear Equation in 2 Variables Ax + By = C Linear Equation in 3 Variables 3x + 4y = 12 Ax + By + Cz = D 3x + 4y + 6z = 12 There is no limit to the number of variables in a linear equation. Animation uses over 100.

What makes it a ‘linear” equation ? If the exponent of the variable(s) is a ‘1’, then it is a linear equation. 3x + 4y + 6z = 12

Solutions of Linear equations What does “solution” mean ? Vocabulary Solution: the number the variable must equal in order to make the statement true. in order to make the statement true. x + 1 = 2 x + 1 = 2 The solution is: x = 1 The solution is: x = 1

“Solving” Linear equations What does it mean to “solve” an equation? Vocabulary Solve: using properties to re-write the equation in the form: x = (some exact value) in the form: x = (some exact value) in order to make the statement true. in order to make the statement true. x + 1 = 2 x + 1 = 2 (subtraction property of equality) (subtraction property of equality) x = 1 x = 1 (solution) (solution)

Distributive Property of Addition over Multiplication 2(x + 4)= (2 * x)+(2 * 4)= 2x + 8 Biggest 2 errors in the distributive property: Trying to multiply when the operation is add or subtract Failing to distribute a negative to BOTH terms inside parentheses 5 - (x - 4)= 5 x – 20 NO NO NO!!! = 5 – x – (x - 4) NO NO NO!!!

Your turn: Solve these equations

Vocabulary Linear Inequality (in one variable): Ax + B < C or Ax + B > C or Ax + B ≤ C or Ax + B ≥ C. 3x – 2 < 4

More Vocabulary Equivalent Inequalities:. Equivalent Inequalities: An inequality that has the same solution as the original inequality. x + 2 = 4 x = 2 x + 2 < x < 2 (subtraction property of inequality) (subtraction property of equality) Equivalent Equations: Equivalent Equations: An equation that has the same solution as the original equation.

Solving inequalities Solving inequalities (variable on both sides of a single inequality symbol) 3x + 1 ≤ 2x + 6 KEY POINT: collect variable on the side that will result in a positive coefficient. -2x x + 1 ≤ 6 x ≤ 5 x ≤ 5

Your turn: Solve the inequality x – 2 < 5x x + 2 ≤ (x – 3 ) ≥ 8

The “Gotcha” of Inequalities 2 – 2x ≤ 6 + 2x 2 ≤ 2x ≤ 2x ÷2 -2 ≤ x 2 – 2x ≤ x ≤ 4 ÷ -2 x ≤ -2 Anytime you multiply or divide by a negative number, you must switch the direction of the inequality !!

Solving inequalities Solving inequalities (variable on both sides of a single inequality symbol) 3x + 1 ≤ 2x + 6 To avoid the “gotcha”: collect the variable on the side that will result in a positive coefficient. -2x x + 1 ≤ 6 x ≤ 5 x ≤ 5

Your turn: Solve the inequality 10. 2(x – 4) < 4x x – 6 ≤ 3 – x x ≥ 9x + 4

Compound inequalities Compound inequalities (two inequality symbols) 5 ≤ x + 1 and 4 ≤ x 4 ≤ x Same as: 4 ≤ x < 8 x + 1 < 9 and x < 8 x < 8 5 ≤ x + 1 < 9

Compound inequalities Compound inequalities (two inequality symbols) 5 ≤ x + 1 < 9 KEY POINT: subtraction property of inequality  do the same thing (left-middle-right) ≤ x < 8 4 ≤ x < 8 4 ≤ x and x < 8 Same as: 4 ≤ x and x < 8

Your turn: Solve the inequality < 4 – x ≤ < x + 1 and x + 1 ≤ 6

Solving inequalities Solving inequalities (“or” type) x - 2 ≤ 3 or x + 2 > 8 KEY POINT: treat “or” type compound inequalities as two separate inequalities. +2 x ≤ 5 x ≤ x > 6 x > 6or x ≤ 5 x ≤ 5 x > 6 x > 6or

Your turn: Solve the inequality x - 7 ≤ 5 or 3x + 2 > x + 1 ≤ -3 or x – 2 > 0

Sometimes there is no solution 2(x – 4) > 2x + 1 Solution: the value(s) of the variable that make the statement true. the statement true. 2x – 8 > 2x x -2x – 8 > 1 – 8 > 1 No solution: when the variable dissappears and the resulting statement is false.

Sometimes the solution is all real numbers. Solution: the value(s) of the variable that make the statement true. 4x – 5 ≤ 4(x + 2) -4x – 5 ≤ 8 Infinitely many solutions: Infinitely many solutions: when the variable dissappears and the resulting statement is true. 4x – 5 ≤ 4x + 8

Your turn: Solve the inequality (3x – 1) > 3(2x + 3) x + 3 ≤ 3(x + 2) – x

Graphing Single Variable inequalities x > What part of the number line is greater than 3 ?

Graphing Single Variable inequalities x < What part of the number line is less than 5 ?

Your turn: Graph the following x ≥ > x

Graphing Compound Inequalities x > 3 and x < What part is x > 3 ? And means both conditions must be met What part is x < 5? What is the intersection or overlap of the two?

Vocabulary x > 3 and x < 5 Hint: Inequality with “and” looks like:   Compound inequality Hint: This can also be written as: 3 < x < 5

Your turn: Graph the following compound inequalities. x > 2 and x < < x ≤ 5

Graphing “or” type compound inequalities. x ≤ 3 or x > 5 Or means: the points that satisfy either condition Which part is x > 5 ?Which part is x ≤ 3 ? Hint: inequality with “OR” looks like:  

Your turn: 21: 21. Solve and graph the compound inequality: x + 3 ≤ 5 or x - 3 > 2

Verbal Inequalities The cost of a car is at most $20,000. It takes Jehah no less than 5 minutes to run a mile. It takes between 3 and 8 months to build a house. The cost of a loaf of bread is less than $2 You can’t buy a car for less than $8000.

Your turn: Your turn: (a) Write in inequality notation (b) Graph the inequality It never gets above 100 degrees in Huntsville. 22.There are least 65,000 spectators at the game You can fit, at most, 5 cars in your garage.

Three Ways to show an Inequality 1. Inequality: x > 3 2. Bracket Notation: (3, ) 3. Number line Notation: x ≤ (, 2]

Inequalities Involving Fractions

Another example?

Your Turn: Solve this inequality 25.

Homework P-3: evens: 2-10, 18-26, 32-44, 54 (18 problems)