Review of Inequalities and Functions

Review of Inequalities and Absolute Value

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Inequalities

• Solutions describe a set of numbers, rather than just one singular number.

• Solutions can be less than (<), greater than (>), less than or equal to ( $\le$ ), or greater than or equal to ( $\ge$ ) a boundary number.

Multiple Choice

When you graph an inequality, you use a closed dot when you use which symbols?

• ≤, ≥

• <, >

• ≤, <

• ≥, >

Multiple Choice

What inequality does the number line graph represent?

• x ≤ -4

• x ≥ -4

• x < -4

• x < 4

Multiple Choice

Match the graph with its inequality.

• b > –2

• b < –2

• b ≥ –2

• b ≤ –2

Multiple Choice

Match the graph with its inequality.

• a > 11

• a < 11

• a ≥ 11

• a ≤ 11

Solving Inequalities

• Use the same Properties of Equality and Algebraic Properties as you would an equation.

• When multiplying or dividing by a negative number, you must switch the direction of the inequality sign.

• Example:

   $-x+3\ge-9$  (original equation)

•  $-x+3-3\ge-9-3$  (Subtraction Property of Equality

•  $-x\ge-12$  (Combine Like Terms)

•  $\frac{-x}{-1}\ge\frac{-12}{-1}$  (Division Property of Equality)

•  $x\le12$  

Multiple Choice

You flip an inequality symbol when you...

• subtract

• multiply and divide only

• multiple by a negative number

• multiple or divide by a negative number

Multiple Choice

-4x + 14 ≤ 54

• x ≥ - 10

• x ≤ -10

• x ≥ 10

• x ≤ 10

Compound Inequalities

• The "AND" inequality:

   $a<x<b$  /  $x>a\ AND\ x<b$  

• The "OR" Inequality: $x<a\ OR\ x>b$  

• AND Inequalities always express the OVERLAP between two inequalities (what numbers do the two inequalities have in common?)

• The Graph for -6 < x < 4 would be all numbers between -6 and 4

• OR Inequalities always express the UNION of two inequalities.

• The Graph for x < -6 or x > 4 would be all numbers to the left of -6 and numbers to the right of 4 graphed on the same number line.

Multiple Choice

What inequality is graphed?

Pay close attention to the signs!

• x > 2 and x ≤ -3

• x > 2 and x < -3

• x > 2 or x ≤ -3

• x > 2 or x < -3

Multiple Choice

What inequality is graphed?

Pay close attention to the signs!

• 2 < x < 4

• 2 ≤ x ≤ 4

• x > 2 or x < 4

• x ≥ 2 or x ≤ 4

Multiple Choice

What inequality is shown?

Pay close attention to the signs!

• x ≥ 0 OR x < 5

• 0 ≤ x ≤ 5

• 0 ≤ x < 5

• 0 ≥ x > 5

Solving Compound Inequalities

• Solving the OR Inequality: Solve BOTH Inequalities separately

• Treat them as if they were normal inequalities and graph both sets of lines based on their rule.

• Solving the AND Inequality: Solve from the INSIDE OUT

• The goal is to isolate x in the middle expression

Example:

 $-4\le3x-4<17$  

•  $-4+4\le3x-4+4<17+4$  (Addition Property of Equality)

•  $0\le3x<21$  (Combine Like Terms)

•  $\frac{0}{3}\le\frac{3x}{3}<\frac{21}{3}$  (Division Property of Equality)

•  $0\le x<7$  

Multiple Choice

Solve:

3x + 2 < -4 or 4x + 4 > 24

• x < -2 or x > 5

• x < 2 or x > 5

• x < -2 and x > 5

• x < 2 and x > 5

Multiple Choice

Solve: -3 ≤ 2x - 1 ≤ 5

• -1 ≤ x ≤ 3

• -2 ≤ x ≤ 6

• -3 ≤ x ≤ 3

• 1 ≤ x ≤ 3

Absolute Value Equations

• Absolute Value --> Distance from Zero

• Absolute Value equations must be equal to a number greater than or equal to zero, otherwise there is no solution to that equation.

• Example: $\left|x+3\right|=7$  

•  $x+3=7,\ x+3=-7$  

•  $x+3-3=7-3,\ x+3-3=-7-3$  (subtraction property of equality)

•  $x=4,\ x=-10$  (combine like terms)

Multiple Choice

Solve the absolute value equation. 

• {7}
• {-7, 7}
• {0, 7}
• No Solution

Multiple Choice

Solve for x
l 5x l = 40

• x = 8 and x =-8
• x = 35 and x = -35
• Only x = 8
• No Solution

Absolute Value Inequalities

• ALWAYS about the signs!

•  $\left|x\right|<a$  results in an AND INEQUALITY (solve inside out,  $-a<x<a$ )

•  $\left|x\right|\ge a$  results in an OR INEQUALITY (solve each separately,  $x<-a\ OR\ x>a$  

Multiple Choice

| x + 1 | ≥ 3

• No Solution
• All Real Numbers
• x ≤ -4  or  x ≥ 2
• -4 ≤ x ≤ 2

That's a WRAP!

That's our first semester of Algebra 1B. Come see me live if you need help! You are ALL important to me!

-Mr. Schulz