Section 5-1: Solving Linear Inequalities by Addition and Subtraction
Inequality –
Addition Property of InequalitiesWords:
Symbols:
Section 5-1: Solving Linear Inequalities by Addition and Subtraction
Inequality – A mathematical sentence that contains < , > , < , or >
Addition Property of InequalitiesWords: Any number is allowed to be added to both sides of a true inequality
Symbols: If a > b, then a + c > b + cIf a < b, then a + c < b + c
Section 5-1: Solving Linear Inequalities by Addition and Subtraction
Graphing inequalities on a number line:1) Put a circle on the _____________ point2) Fill the circle in if the sign is _______ or _______3) Leave the circle open if the sign is _____ or ____4) Shade the left side if the variable is
____________ to the number5) Shade the right side if the variable is
____________ to the numberEx) Graph x < -9 Ex) Graph 4 < x
Section 5-1: Solving Linear Inequalities by Addition and Subtraction
Solve the following inequalities and graph the solution set on a number line.1) x – 12 > 8 2) 22 > x – 8 3) x – 14 < -19
Section 5-1: Solving Linear Inequalities by Addition and Subtraction
Subtraction Property of InequalitiesWords:
Symbols:
Section 5-1: Solving Linear Inequalities by Addition and Subtraction
Subtraction Property of InequalitiesWords: Any number is allowed to be subtracted from both sides of a true inequality
Symbols: If a > b, then a – c > b – cIf a < b, then a – c < b – c
Section 5-1: Solving Linear Inequalities by Addition and Subtraction
Solve the following inequalities and graph the solution set on a number line.1) x + 19 > 56 2) 18 > x + 8 3) 22 + x < 5
Section 5-1: Solving Linear Inequalities by Addition and Subtraction
Solve the following inequalities and graph the solution set on a number line.1) 3x + 6 < 4x 2) 10x < 9n – 1
Section 5-2: Solving Inequalities by Multiplication and Division
Multiplication Property of Inequalities: Part 1Words:
Symbols:
Section 5-2: Solving Inequalities by Multiplication and Division
Multiplication Property of Inequalities: Part 1Words: Any positive number is allowed to be multiplied to both sides of an inequality
Symbols (c > 0):If a > b, then ac > bcIf a < b, then ac < bc
Section 5-2: Solving Inequalities by Multiplication and Division
Multiplication Property of Inequalities: Part 2Words:
Symbols:
Section 5-2: Solving Inequalities by Multiplication and Division
Multiplication Property of Inequalities: Part 2Words: Any negative number is allowed to be multiplied to both sides of an inequality, as long as the sign gets flipped!
Symbols (c < 0):If a > b, then ac < bcIf a < b, then ac > bc
Section 5-2: Solving Inequalities by Multiplication and Division
Division Property of Inequalities: Part 1Words: Any positive number is allowed to be divided to both sides of an inequality
Symbols (c > 0): If a > b, then
If a < b, then
Section 5-2: Solving Inequalities by Multiplication and Division
Division Property of Inequalities: Part 2Words: Any negative number is allowed to be divided to both sides of an inequality, as long as the sign gets flipped!Symbols (c < 0): If a > b, then
If a < b, then
Section 5-2: Solving Inequalities by Multiplication and Division
Solve and graph the solution set1) 4x > 16 2) -7x < 147
3) -15 < 5x 4) -20 > -10x
Section 5-3: Solving Multi-Step Inequalities
Solve the following multi-step inequalities. Graph the solution set.1) -11x – 13 > 42 2) 15 + 2x < 31
3) 23 > 10 – 2x
Section 5-3: Solving Multi-Step Inequalities
Solve the following multi-step inequalities. Graph the solution set.1) 4(3x – 5) + 7 > 8x + 3
2) 2(x + 6) > -3(8 – x)
Section 5-3: Solving Multi-Step Inequalities
Translate the verbal phrase into an expression and solveFive minus 6 times a number n is more than four times the number plus 45
Section 5-3: Solving Multi-Step Inequalities, SPECIAL SOLUTIONS
Solve the following inequalities. Indicate if there are no solutions or all real number solutions.1) 9x – 5(x – 5) < 4(x – 3) 2) 3(4x + 6) < 42 + 6(2x – 4)
Day 1, Section 5-4: Compound Inequalities
Compound Inequality – two inequalities combined into one with an overlapping solution set
And vs. OrGraph Graph
Inequality Inequality
Intersection Union
Day 1, Section 5-4: Compound Inequalities (ANDs)
Ex) Graph the intersection of the two inequalities and write a compound inequality for
x > 3 and x < 7
Ex) Solve and graph -2 < x – 3 < 4
Day 1, Section 5-4: Compound Inequalities (ANDs)
Solve the following inequalities and graph the solution set on a number lineEx) y – 3 > -11 and y – 3 < -8
Ex) 6 < r + 7 < 10
Day 2, Section 5-4: Compound Inequalities (ORs)
Graph the following inequalities on the same number line:
x > 2 or x < -1
Ex) Solve and graph the solution set for: -2x + 7 < 13 or 5x + 12 > 37
Day 2, Section 5-4: Compound Inequalities (ORs)
Solve the following inequalities and graph the solution set
1) a + 1 < 4 2) x < 9 or 2 + 4x < 10
Or a – 1 > 3
Day 1, Section 5-5 Absolute Value Inequalities (distance is “less than”
What does mean?
Absolute value inequalities that have _______ or _________ symbols are treated like _______ problems!! The solution set will be an _____________.
Day 1, Section 5-5 Absolute Value Inequalities (distance is “less than”)
What does mean?The distance from zero on the number line is less than 3.
Absolute value inequalities that have less than (<) or less than or equal (<) to symbols are treated like AND problems!! The solution set will be an intersection.
5-4: Steps for Solving Abs. Value Inequalities: or 1) Re-write as an AND problem with boundaries +c and –c. (Lose the abs. value symbols)
2) Solve the AND problem
3) Graph the solution
Ex)
Day 1, Section 5-5 Absolute Value Inequalities (distance is “less than”)
Solve the following absolute value problems. Graph the solution set.1)
2)
3)4)
Day 2, Section 5-5 Absolute Value Inequalities(distance is “greater than”)
What does mean?
Absolute value inequalities that have _______ or _________ symbols are treated like _______ problems!! The solution set will be a _____________.
Day 2, Section 5-5 Absolute Value Inequalities(distance is “greater than”)
What does mean?The distance from 0 on the number line is greater than 3
Absolute value inequalities that have greater than (>) or greater than or equal to (>) symbols are treated like OR problems!! The solution set will be a union.
5-4: Steps for Solving Abs. Value Inequalities: or 1) Re-write as an OR problem. Split into 2 problems, one with +c and one with –c (Lose the abs. value symbols)***YOU MUST FLIP SIGN ON –C PROBLEM2) Solve the two problems
3) Graph the solution
Ex)
Day 2, Section 5-5 Absolute Value Inequalities(distance is “greater than”)
Solve the following absolute value inequalities1)
2)
3)4)
Section 5-6: Graphing Linear Inequalities
Linear Equations vs.• Plot any 2 points• Draw a solid line through
the 2 points
Linear Inequalities• Put in slope-int form first• Plot any two points• Determine whether to
connect the points with a solid OR dashed line
• Shade ONE side of the line– You must determine which
side– You shade the solution set!
Section 5-6: Graphing Linear Inequalities
Steps for graphing linear inequalities:1) Plot 2 points on the line2) If the symbol is < or >
connect with a dashed line3) If the symbol is < or >,
connect with a solid line4) Shade above the line for
y > mx + b (or >)5) Shade below the line for
y < mx + b (or <)
Ex. Graph y < 2x – 4