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**Equations and Inequalities**

Chapter 4 Equations and Inequalities

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**4-1-C: Explore – Solve equations using algebra tiles**

Look at this picture. The paper bag on the left contains a certain number of orange blocks. Assuming that the paper bag weighs nothing, can you determines how many blocks are in the orange bag? What do you think we could do? One way is to remove four orange blocks from each side of the balance. The bag contains three blocks. In algebra, an EQUATION is like a balance. The weights on each side of the scale are equal. You can use algebra tiles to model and solve equations.

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**4-1-C: Explore – Solve equations using algebra tiles**

Solve 𝑥+4=7 using algebra tiles. Model the equation. Remove the same number of 1-tiles from each side of the mat so that the variable is by itself on one side. The number of 1 tiles remaining on the right side of the mat represents the value of x. So, x = 3. 1 1 1 1 1 1 x 1 1 1 1 1

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**4-1-C: Explore – Solve equations using algebra tiles**

A zero pair is a number paired with its opposite, like 2 and -2. You can add or subtract a zero pair from either side of an equation without changing its value, because the value of the zero pair is zero. Solve 𝑥+2=−1 with algebra tiles. Model the equation. Add two negative tiles to the left side of the mat and two negative tiles to the right side of the math to form zero pairs on the left. Remove all of the zero pairs from the left side. There are 3 negative tiles on the right side of the mat. So, x = -3. Now, with your partner, work with your tiles to complete book page 207. #1-12. Be sure to write in COMPLETE SENTENCES when answering #9-12.

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**4-1-D: Solve one-step addition and subtraction equations**

An EQUATION is a sentence stating two quantities are equal. We can solve equations using INVERSE OPERATIONS. An inverse operation is an operation used to UNDO another operation. What is the inverse operation of addition? What is the inverse operation of multiplication? SUBTRACTION PROPERTY OF EQUALITY If you subtract the same number from each side of an equation, the two sides remain equal. Symbols: If 𝑎=𝑏, 𝑡ℎ𝑒𝑛 𝑎−𝑐=𝑏−𝑐. Examples: 6= 6 𝑥+2= 6 −2=− −2=−2 4= 4 𝑥 =4

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**4-1-D: Solve one-step addition and subtraction equations**

Let’s look at three more examples before we try it on our own. 𝑦+6=9 𝑥+3=1 −3=𝑎+4 −6=−6 −3=−3 −4= −4 𝑦 =3 𝑥 =−2 −7=𝑎 Now you try! 𝑎+3=10 9=𝑟+2 𝑥+8=5 𝑦+15=11 If you are ever stuck and do not know what to do, ask yourself these three questions: What is my variable? What is being done to my variable? How can I UNDO what is being done to my variable? a = 7 r = 7 x = -3 y = -4

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**4-1-D: Solve one-step addition and subtraction equations**

ADDITION PROPERTY OF EQUALITY If you add the same number to each side of an equation, the two sides remain equal. Symbols: If 𝑎=𝑏, 𝑡ℎ𝑒𝑛 𝑎+𝑐=𝑏+𝑐. Examples: 5= 5 𝑥−2= 4 +3= =+2 8= 8 𝑥 =6 Let’s look at a few more examples: 𝑦−3=4 𝑟−4=−2 𝑞−8=−9 +3= = =+8 𝑦 =7 𝑟 =2 𝑞 =−1

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**4-1-D: Solve one-step addition and subtraction equations**

Now you try! 𝑠−8=9 𝑤−7=11 −1=𝑞−8 −2=𝑝−13 REAL WORLD Examples The highest recorded temperatures in Warsaw, Missouri, is 118°F. This is 158° greater than the lowest recorded temperature. Write and solve an equation to find the lowest recorded temperature. 118=158+𝑡 The average span of a tiger is 17 years. This is 3 years less than the average lifespan of a lion. Write and solve an equation to find the average lifespan of a lion. 𝑙−3=17 s = 17 w = 18 x = 7 y = 11 On your own, practice solving addition and subtraction equations by completing #1 - 8 on page 211 in your math book.

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**4-2-b: Solve one step multiplication and division equations**

The expression 3𝑥 means 3 times the value of x. The numerical factor of a multiplication expression like 3x is called a coefficient. (In other words, the coefficient is the number attached to a variable.) So 3 is the coefficient of 3x. Sometimes, you can use a drawing to represent an equation. Look at 3𝑥=12. This means “three groups of a something equal twelve. Start with three groups and then evenly divide up the twelve. COUNT WITH ME! So x = 4. Draw a picture for these: 8=2𝑥 2𝑥=−8 3𝑥=−9

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**4-2-b: Solve one step multiplication and division equations**

DIVISION PROPERTY OF EQUALITY If you divide each side of an equation by the same nonzero number, then two sides remain equal. Symbols: If 𝑎=𝑏, 𝑎𝑛𝑑 𝑐≠0, 𝑡ℎ𝑒𝑛 𝑎 𝑐 = 𝑏 𝑐 . Examples: 8=8 2𝑥=−6 8 2 = 𝑥 2 = −6 2 4=4 𝑥=−3 Remember your three questions: What is my variable? What is being done to my variable? How can I UNDO what is being done to my variable?

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**4-2-b: Solve one step multiplication and division equations**

Check out a few more examples: 30=6𝑥 −6𝑎=36 −9𝑑=−72 30 6 = 6𝑥 6 −6𝑎 −6 = 36 −6 −9𝑑 −9 = −72 −9 𝟓=𝒙 𝒂=−𝟔 𝒅=𝟖 Now try some on your own. 9𝑒=27 −4𝑗=36 −12𝑦=60 −4𝑠=−16 e = 3 j = -9 x = -5 y = 4

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**4-2-b: Solve one step multiplication and division equations**

MULTIPLICATION PROPERTY OF EQUALITY If you multiply each side of an equation by the same number, then two sides remain equal. Symbols: If 𝑎=𝑏, 𝑡ℎ𝑒𝑛 𝑎𝑐=𝑏𝑐. Examples: = 𝑥 4 =7 3 −6 =3(−6) 𝑥 4 (4)=7(4) −18=− 𝑥=28 Check out a few examples before you try some on your own.. 𝑦 −3 =− 𝑚 5 =− = 𝑏 −6 𝑦 −3 (−3)=−8(−3) 𝑚 5 (5)=−7(5) 30(−6)= 𝑏 −6 (−6) 𝑦= 𝑚=− −180=𝑏

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**4-2-b: Solve one step multiplication and division equations**

Now try to solve some on your own! 𝑚 10 =7 ℎ −3 =12 −30= 𝑞 −5 −8= 𝑐 −10 REAL WORLD EXAMPLES Mrs. Bybee’s car can travel an average of 24 miles on each gallon of gasoline. Write and solve an equation to find how many gallons of gasoline she will need for a trip of 348 miles. 24𝑥=348 A Fitch ferret has a mass of about 2 kilograms. To find its weight in pounds p, you can use the formula 𝑝 2 =2.2. How many pounds does a Fitch ferret weigh? A FORMULA is an equation that shows the relationship among certain quantities. m = 70 h = -36 x =150 y = 80 On your own, practice solving multiplication and division equations by completing #1 - 8 on page 218 in your math book.

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**4-2-d: Solve equations with rational coefficients**

Mr. Hoffmann rides his bike for hour after school. Then he rides his bike for another hour before dinner. In all, he spends 2∙ 1 2 or 1 hour riding his bike. Copy and complete the table below. What do you notice about the table? Two numbers with a product of 1 are called MULTIPLICATIVE INVERSES, or RECIPROCALS. INVERSE PROPERTY OF MULTIPLICATION The product of a number and its multiplicative inverse is 1. 3 2 × 2 3 = 1 5 × =1 5 6 × 6 5 = 7 8 × 8 7 = × 5 7 =1 2 6 × 6 2 = 7 1 × =1 ×8=1

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**4-2-d: Solve equations with rational coefficients**

Find the multiplicative inverse of the following numbers: Sometimes the coefficient of a term in a multiplication problem is a rational number. If the coefficient is a decimal, divide each side by the coefficient. 0.75𝑥=3.75 0.75𝑥 0.75 = 𝑥=5 If the coefficient is a fraction, multiply each side of the equation by the reciprocal of the coefficient. 2 3 𝑥=4 𝑥=4 3 2 𝑥=6

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**4-2-d: Solve equations with rational coefficients**

Let’s practice with both decimal and fraction coefficients! 1.2𝑥=6 2 5 𝑡= 12 25 2.8=0.4𝑑 24 16 = 3 4 𝑎 −2.4𝑏=14.4 − 1 2 𝑔=−3 1 3 3.6ℎ=−10.8 3 5 𝑐=−6 1 4 REAL WORLD EXAMPLES Suppose the ice cream cones cost $2.80 each. How many ice cream cones could Miss Holloway buy with $42? Wilson has 9 pounds of trail mix. How many pound bags of trail mix can he make? x = 5 t = 𝑜𝑟 1 1 5 d =7 a = 2 b = -6 g = 6 2 3 h = -3 c = − On your own, practice solving equations with rational coefficients by completing # on page 224 in your math book.

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**4-3-b: Solve two step equations**

A TWO-STEP EQUATION has two different operations. To solve a two-step equation, UNDO the operations in REVERSE order of the ORDER of OPERATIONS. 2𝑥+4=10 −4=−4 2𝑥=6 2𝑥 2 = 6 2 𝑥=3 In order of operations, the multiplication would be evaluated first. But because we are doing things in the reverse order, we must UNDO the addition first. After the first step, we’re left with a one-step division equation!

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**4-3-b: Solve two step equations**

Here are a few more examples before you try some on your own. 4𝑥+5=13 −5𝑠+8=−2 − 𝑤=10 −5=− −8=− =+2 4𝑥= −5𝑠=− 𝑤=12 4𝑥 4 = −5𝑠 −5 = −10 − 𝑤= 𝒙=𝟐 𝒔=𝟐 𝒘=𝟏𝟖 Key Concept Step 1: Undo the addition or subtraction first. Step 2: Then undo the multiplication or division. STEP 1 STEP 2

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**4-3-b: Solve two step equations**

Now you try! 3𝑥+1=10 −4𝑦−4=8 6𝑚+1=−23 2𝑟−3.1=1.7 −8𝑠+1=33 16=0.5𝑞−8 5+4𝑑=37 𝑝=8 REAL WORLD EXAMPLE A fitness club is having a special offer where you pay $22 to join plus a $16 monthly fee. You have $150 to spend. Write and solve an equation to find how many months you can use the fitness club. 22+16𝑚=150 x = 3 y = - 3 m = - 4 r = 2.4 s = - 4 q = 48 d = 8 p = - 9 On your own, practice solving two-step equations by completing #1 - 7 on page 232 in your math book.

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**4-3-d: Solve equations with variables on both sides**

As you discovered in the Geology Rocks Equations activity, you can use the properties of equality with variables on each side. The new equation is equivalent to the original equation. Remember the four properties of equality: Subtraction Property of Equality Addition Property of Equality Division Property of Equality Multiplication Property of Equality Key Concepts: We have to use the equality properties in order to get the variable on to ONE side of the equation. We can only solve the equation if we can ISOLATE the variable. Remember that what you do to one side of the equation, you MUST do the EXACT SAME THING to the other side of the equation.

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**4-3-d: Solve equations with variables on both sides**

Let’s look at some examples 𝟐𝒙+𝟒=𝒙+𝟔 −𝑥 =−𝑥 𝑥+4=6 −4 =−4 𝒙=𝟐 TIP: Work with the variables first. Once you can get those to one side, then you’re left with two-step equations. SIGNS: Sometimes, equations will be written differently. A good strategy is to always look at the sign in front of numbers. 𝟑𝒙−𝟒=𝒙−𝟔 −𝑥 =−𝑥 2𝑥−4=−6 +4=+4 2𝑥=−2 2𝑥 2 = −2 2 𝒙=−𝟏 𝟗−𝟒𝒙=𝟐𝒙+𝟔 𝑥=+4𝑥 =6𝑥+6 −6= −6 3=6𝑥 3 6 = 6𝑥 6 𝟏 𝟐 =𝒙

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**4-3-d: Solve equations with variables on both sides**

It does not matter which side your variable ends up on. When making your first move, ask yourself, which variable would be easier to move. For example, look at the equation 9−4𝑥=2𝑥+6. Would it be easier to ADD 4x to both sides? Or would it be easier to SUBRACT 2x from both sides? In the end, it does NOT matter which you do, you will end up with the same answer either way. Now you try! 5𝑥+1=7𝑥−11 2𝑥−27=45−𝑥 18.6−2𝑥=3𝑥−2.4 −7+𝑥= 𝑥 10 −16 −1−5𝑥=15𝑥−6 −20−2𝑥=2𝑥+4 x = 6 x = 24 x = 4.2 x = - 10 x = ¼ x = -6

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**4-3-d: Solve equations with variables on both sides**

REAL WORLD EXAMPLES Bill averages 10 points a game and has 110 points for the season. Aaron averages 12 points a game and has 96 points for the season. Write and solve an equation to find how many games it will take until they tie in points scored if they continue at the same rate. Remember to ask yourself: What stays constant no matter what? What is dependent upon the number of games? We know that Bill already has 110 points for the season – so that’s constant. He averages 10 points PER game, so we know we’ll multiply 10 by the number of games, g. We know that Aaron already has 96 points – so that’s constant. He averages 12 points a game, so we know we’ll multiply 12 by the number of games, g. Since we want to find the SAME g for both players, we can set these equal to each other and solve. 110+10𝑔=96+12𝑔 On your own, practice solving equations by completing numbers 1-10 on page 238 in your math book.

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**4-4-b: Solve inequalities by addition or subtraction**

An INEQUALITY is a mathematical sentence that compares two quantities. <,≤ - LESS THAN, and LESS THAN OR EQUAL TO (Looks like a crooked L) >,≥ - GREATER THAN, and GREATER THAN OR EQUAL TO When we are solving inequalities, we are finding values that make the inequality true. We want to identify ALL the numbers that make then inequality true. KEY CONCEPT When you add or subtract the same number from each side of an inequality, the inequality remains true. Sound familiar? Inequalities are solved using the same properties of equality that we use for equation.

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**4-4-b: Solve inequalities by addition or subtraction**

Words is less than is fewer than is greater than is more than exceeds is less than or equal to is no more than is at most is greater than or equal to is no less than is at least Symbols < > ≤ ≥ Check out a few examples! −3<𝑛−8 5<𝑛 𝑦−6≥−13 𝑦≥−7 𝑥+0. 7<−0.3 − −0.7 𝑥<−1

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**4-4-b: Solve inequalities by addition or subtraction**

Once we solve inequalities, we also have to graph the solution on a number line. OPEN AND CLOSED CIRCLES When graphing inequalities, an open circle is used when the value should not be included in the solution, as with > and < inequalities. A closed circle indicates the value is included in the solution, as with ≥ and ≤ inequalities. Let’s practice graphing our answers from the previous examples. 5<𝑛, or 𝑛>5 Notice that either way we write the inequality, the n is still greater. 𝑦≥−7 𝑥<−1

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**4-4-b: Solve inequalities by addition or subtraction**

Now you try! Solve each inequality. Graph the solution set on a number line. −11>𝑡+7 𝑚+5≥−1 −21<𝑎−16 𝑡−6.2<4 𝑢− 1 5 ≤ 3 10 6>𝑥+3 1 3 Write an inequality and solve each problem Four more than a number is more than 13. The sum of a number and 19 is at least 8. Eight less than a number is less than 10. The difference between a number and 21 is no more than 14. −18>𝑡 or 𝑡<−18 𝑚≥−6 −5<𝑎 or 𝑎>−5 𝑡<10.2 𝑛≤ 1 2 2 2 3 >𝑥 or 𝑥<2 2 3

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**4-4-c: Solve inequalities by multiplication and division**

Similar to equations, we will also use the Division and Multiplication Properties of Equality to solve inequalities. What you do to one side, you must to do the other… Key Concept When you multiply or divide each side of an inequality by a positive number, the inequality remains true. 4𝑥<40 4𝑥 4 < 40 4 𝑥<10 When you multiply or divide each side of an inequality by a negative number, the inequality sign must be reversed (or flipped) for the inequality to remain true. 𝑘 −2 >9 −2 𝑘 −2 <9(−2) 𝑘<−18

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**4-4-c: Solve inequalities by multiplication and division**

Check out a few more examples: Now try some on your own! 4𝑥≥36 20<5𝑡 𝑠 −6 >−16 ℎ 9 >9 −7𝑦<35 −10𝑛>−20 𝑡 −2 <6 FLIPPED 𝟔𝒚<𝟏𝟖 6𝑦 6 < 18 6 𝑦<3 −𝟑𝒔≥𝟑𝟑 −3𝑠 −3 ≤ 33 −3 𝑠≤−11 𝟔𝟎≤ 𝒎 𝟑 3 60≤ 𝑚 3 (3) 180≤𝑚 or 𝑚≥180 𝒘 −𝟓 ≤𝟗 −5 𝑤 −5 ≥9 −5 𝑤≥−45 FLIPPED 𝑥≥9 4<𝑡 or 𝑡>4 𝑠<96 ℎ>81 𝑦>−5 𝑛<2 𝑡>−12

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**Solving 2-Step inequalities**

Two-step inequalities are solve similarly to 2-step equations. Remember: Undo operations in the reverse order of Order of Operations Remember to FLIP the inequality sign if you multiply or divide by a negative number. Solve each inequality. Graph the solution on a number line. −4−5𝑣<−29 −2+ 𝑟 9 >−1 −1+4𝑥≤31 𝑚 6 −5≥−6 3𝑛−3>12 9(𝑥+7)≥198 −40<5+5𝑥 −6> 𝑚 2 −4 v > 5 r > 9 x ≤ 8 m ≥ -6 n > 5 x ≥ 15 x > -9 m < -4

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