Unit: Expressions, Equations, & inequalities

Intro to combining like terms • Say we have 2x and we add it to 3x. we don't know the value of x but whatever that value is we could add it to itself so we could write this as (x + x) + (x + x + x) so we have 5x. All we did was add the two numbers that were multiplying the x. these numbers are called coefficients which are constant numbers that are multiplied by the variable. •If we were to add (7p + 5r + 2p) then we wouldn't add p to r instead we express them individually so we have (5r + 9p)

Combining like terms with negative coefficients & distribution •Say we want to simplify the expression 2(3x + 5) we would write this using the distributive property by distributing the 2 so 2(3x) + 2(5) which is (6x + 10) •7(3y - 5) - 2(10 + 4y) we would start at the left-hand side by distributing the 7. so we would have (21y) & (-35) and then the right-hand side so we could think of the 2 in - 2(10 + 4y) as negative 2 so we would distribute -2 which would give us (-20) & (-8y) now to simplify we would subtract 21y from 8y = (13y) then we subtract -35 from -20 = (55) these two expressions as a whole would be (13y) & (-55)

combining like terms with negative coefficients • simplify (-3y + 4xy - 2x² + 2x + y² - 4xy + 2y + 3x²) note that a y is different from an XY, y², so even if you see the same letter they ARE NOT the same. so first we will do the y terms which can be written as (-3y + 2y) then the XY terms. + (4xy - 4xy) them the x². - (-2x² + 3x²) then the x terms. + (2x) then the y². + (y²) so this would be written as (-y + x² + 2x + y²) •combine like terms to create an equivalent expression: -4p + (-2) + 2p + 3) first combine the numeric terms: (-2p -2 + 3) = (-2p + 1) •Simplify to create an equivalent expression: (-k + 2(-2k - 5). distribute the 2 to each term inside the parentheses: -k + 2(-2k -5) = -k -4k -10. Combine the k-terms: -5k - 10. •simplify to create an equivalent expression: 7n - (4n - 3) The minus sign in front of the parentheses means we multiply each term inside the parentheses by -1: 7n -1(4n - 3) = 7n - 4n + 3) then combine the n terms: (3n + 3)

Combining like terms with rational coefficients •simplify (-5.55) - (8.55c) + (4.35c) So first combine the 'c' terms. so (-8.55 + 4.35)c = (-4.2c) so then we would combine the terms (-5.55) - (4.2c) •Simplfy (2/5m) - (4/5) - (3.5m) first do the m terms (2/5) - (3/5)m = (-1/5m) then combine the terms (-1/5m) - (4/5) •Simplify 2(1/5m - 2/5) + (3/5) first distribute the 2 so (2 • 1/5)m + (2 • -2/5)m = (2/5m) then combine the terms (2/5m) + (-1/5)

The distributive property with variables • Let's say 12 for the example. we could say 12 is the product of 2 and 6. We could also say 2 and 6 are factors of 12 because if you take the product of them you get 12. eg (2 • 6 = 12). you could also say (2 • 6) is 12 in its factored form. • You could also do prime factorization by breaking it up by its prime factors which are ( 2 • 2 • 3) which are the prime factors of (12) • let's try the expression (2 + 4x) we could break this up as 2(1 + 2x) = (2 + 4x) this is its factored form or we factored out the 2 by breaking it up into 2 of its factors. • Say we have 6x + 30. we want to figure out if we could break up two of these forms so they have a common factor. we could break this up as (6 • x + 6 • 5) we are doing the reverse of distributive property by taking out the 6 which gives us 6(x + 5) • Say we have 1/2 - 3/2x. we want to write this as a factored form by factoring out something. we could factor out 1/2 by writing it as 1/2 • 1 - 1/2(3x) = 1/2 (1 - 3x) • We could also do ax + ay. Both of these are products of a so we could factor out the a by writing it as an (x + y)

Equivalent expressions: Negative numbers & distribution • which following expressions are equivalent to- 2(-6c + 3) + 4c? •(-8c + 6) or (3(-4c + 2) + 4c = -8c + 6) The answer is both because using the distributive property the original expression could be written as = -12c + 6 + 4c = -8c + 6. • Which of the following expression are equivalent to (-6n + (-12) + 4n) which equals (-2 - 12)? •4(n -3) - 6n) = (-2n - 12) or 2 (2n - 6) which equals (4n - 12) So it's the first one.

Interpreting Linear expressions •Jack Is buying a pony. the price of the pony is P dollars, and jack also pays a 25% tax. So the price before tax is P. The tax is (0.25 P). and the total bill is (P + 0.25 P) or (1.25 P)

Writing expressions and word problems • The price of a visit to a dentist is $50. If the dentist fills any cavities, an additional charge of $100 per cavity gets added to the bill. if the dentist fills n cavities, what will the cost of the visit be? write the answer as an expression. (50 + 100n) because 100n is 100 • n so if you have zero n then you pay zero extra. • Sunny earns $12 per hour. She worked x hours this week. Sadly she was charged $15 for late delivery on Tuesday. How much did sunny earn this week? Write the answer as an expression. (12x - 15)

Two-step equations intro •Same thing to both sides of equations: Say we have a scale that is balanced on the right side we have 10 blocks of one kg mass, and on the left, we have 3 blocks of one kg mass and one huge block with an unknown mass known as x. both sides are leveled, what is the mass of x? it is 7 because if we removed all three of the blocks on the left and three on the right, leaving us with 7, it would still be balanced. •Two-step Equations intuition: Say (3x + 5 = 17) what is the value of x? first subtract 5 from 17, which is 12, then solve what 3 times x equals 12, which is 4, so )x = 4). •Worled example: two-step equations: solve (-16 = x/4 + 2)

two-step equations with decimals and fractions •Solve (-1⁄3 = j⁄4 - 10⁄3) one way to do this is to isolate the variable 'j' on one side. it's on the right-hand side so we want to get all the things that involve j on the right-hand side and get rid of everything else on the right-hand side. so to get rid of -10⁄3 is by adding ( 10⁄3) but we can't do this to just one side though because it won't be equal anymore, meaning for the right side to be equal to the left we have to do the same thing to both sides so we have to do it to both sides so > (-1⁄3 + 10⁄3 = j⁄4 - 10⁄3 + 10⁄3) which is (3 = j⁄4) to solve for j we could multiply both sides by four because if divide something by four and multiply by four were going to be left with the answer. so (4 • 3 = j⁄4 • 4) which is (12 = j) so j is equal to 12. to verify you could take to the original expression and substitute 12 for j. •Solve (n⁄5 + 0.6 = 2) so isolate the variable by getting rid of 0.6 by subtracting 0.6 from both sides which gives you (n⁄5 = 1.4) then to solve for n multiply both sides by 5 which is (n = 7) so n is equal to 7. •Solve (0.5 (r + 2.75) = 3) we could distribute 0.5 on the left-hand side but another way to do it is by dividing both sides by 0.5 which is (r + 2.75 = 6) and the left is like that because if we distributed the 0.5 on the left side and then divided it by 0.5 it would still be (r + 2.75). isolate the r by subtracting the 2.75 on both sides which are (r = 3.25) so r is equal to 3.25. • Solve ( -4.5 = -0.5(x - 7.1) ) first divide both sides by -0.5 and then add to get x by itself. so (-4.5/-0.5 = -0.5(x - 7.1)/-0.5) which equals (9 = x - 7.1). Next, add 7.1 to both sides with ( 16.1 = x) • Solve (3⁄2 = 5d - ½) first add and then divide to get d by itself. so add 1/2 to both sides which is ( 2 = 5d) then divide both sides by 5 to get d by itself which is (2⁄5 = d) •CHEAT SHEET-- if a number is like this together (-6e) make sure to divide instead of subtracting -6 or adding.