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Exercises 1 The phrase greater than or equal to is symbolically represented by the symbol ≥. We can write "a number z minus 6" as ​A number z minus 6   z − 6.​ Writing a complete inequality, we have z−6≥11.​
Exercises 2 In this exercise, we are asked to translate a statement into symbolic form. To do this, we will break the given statement into pieces. The given statement is ​Twelve is no more than the sumof−1.5 times a number w and 4.​ No more than means less than or equal to. Thus, we so far have 12≤… -1.5 times a number w means we should multiply. Symbolically this gives -1.5w. Our inequality is now 12≤-1.5w… The sum of -1.5 times a number w and 4 means that we should add -1.5w and 4. Combining everything gives the following inequality. 12≤-1.5w+4
Exercises 3 Since the circle at 0 is open, the graph describes all values less than but not equal to 0. In other words, we have a strict inequality and should exclude 0. If we let x represent these values, we can write the following inequality. x<0
Exercises 4 Since the circle at 8 is closed, the graph describes all values greater than or equal to 8. In other words, we have an inequality which is not strict and should include 8. If we let x represent these values, we can write the following inequality. x≥8
Exercises 5 Inequalities can be solved in the same way as equations, by performing inverse operations on both sides until the variable is isolated. The only difference is that when you divide or multiply by a negative number, you must reverse the inequality sign. 9+q≤15LHS−9≤RHS−9q≤6 This inequality tells us that all values less than or equal 6 will satisfy the inequality. Notice that q can equal 6, which we show with a closed dot on the number line.
Exercises 6 Inequalities can be solved in the same way as equations, by performing inverse operations on both sides until the variable is isolated. The only difference is that when you divide or multiply by a negative number, you must reverse the inequality sign. z−(-7)<5a−(-b)=a+bz+7<5LHS−7<RHS−7z<-2 This inequality tells us that all values less than -2 will satisfy the inequality. Notice that z cannot equal -2, which we show with an open dot on the number line.
Exercises 7 Inequalities can be solved in the same way as equations, by performing inverse operations on both sides until the variable is isolated. The only difference is that when you divide or multiply by a negative number, you must reverse the inequality sign. -3<y−4 LHS+4<RHS+4 LHS+4<RHS+4-3+4<y−4+4Add terms 1<yRearrange inequalityy>1 This inequality tells us that all values greater than 1 will satisfy the inequality. Notice that y cannot equal 1, which we show with an open dot on the number line.
Exercises 8 Inequalities can be solved in the same way as equations, by performing inverse operations on both sides until the variable is isolated. The only difference is that, when you divide or multiply by a negative number, you must reverse the inequality sign. 3p≥18LHS/3≥RHS/3p≥6 This inequality tells us that all values greater than or equal to 6 will satisfy the inequality. Notice that p can equal 6, which we show with a closed circle on the number line.
Exercises 9 Inequalities can be solved in the same way as equations, by performing inverse operations on both sides until the variable is isolated. The only difference is that, when you divide or multiply by a negative number, you must reverse the inequality sign. 6>-2w​Multiply by -2 and flip inequality sign-12<wRearrange inequalityw>-12 This inequality tells us that all values greater than -12 will satisfy the inequality. Notice that w cannot equal -12, which we show with an open circle on the number line.
Exercises 10 Inequalities can be solved in the same way as equations. By dividing both sides by -20, we can isolate x. Because we are dividing the inequality by a negative number, we have to reverse the inequality sign. -20x>5Divide by -20 and flip inequality signx<-205​ba​=b/5a/5​x<-41​ This inequality tells us that all values less than -41​ will satisfy the inequality. Notice that x cannot equal -41​, which we show with an open circle on the number line.
Exercises 11 Remember that solving inequalities is done in the same way as solving equations, using inverse operations. Just be careful to flip the inequality symbol when multiplying or dividing the inequality by a negative number. We need to isolate y to solve the inequality. 3y−7≥17LHS+7≥RHS+73y≥24LHS/3≤RHS/3y≥8
Exercises 12 Inequalities are solved in the same way as equations, by isolating the variable. To begin solving the given inequality, we should distribute on both sides of the inequality and then move the variable terms to one side. 8(3g−2)≤12(2g+1)Distribute 824g−16≤12(2g+1)Distribute 1224g−16≤24g+24LHS−24g≤RHS−24g-16≤24 The statement -16≤24 is always true. Thus, the solution to the inequality is all real numbers.
Exercises 13 Remember that solving inequalities is done in the same way as solving equations, we just need to be careful to flip the inequality symbol if we multiply or divide the inequality by a negative number. To begin solving the given inequality, we should distribute on both sides of the inequality and then isolate the variable terms on one side. 6(2x−1)≥3(4x+1)Distribute 612x−6≥3(4x+1)Distribute 312x−6≥12x+3LHS−12x≥RHS−12x-6≱3 The statement -6≥3 is always false. Thus, the inequality has no solution.
Exercises 14 aTo begin we will define three different variables as follows.Let s represent the distance a person can swim. Let t be the number of minutes a person can tread water. Let u be the distance a person can swim underwater without taking a breath. We will separate each of the inequalities into individual sections.Swim at least 100 yards The first requirement is that the person swims at least 100 yards. The phrase at least is another way to say greater than or equal to. Thus, we can write the following inequality to represent this requirement. s≥100 To graph this inequality, we will place a closed circle at 100, because s can equal 100, and shade the region to the right on the number line.Tread water for at least five minutes The second requirement is that the person treads water for at least five minutes. Again, the phrase at least is another way to say greater than or equal to. Thus, we can use the following inequality to represent this requirement. t≥5 To graph this inequality, we will place a closed circle at 5, because t can equal 5, and shade the region to the right on the number line.Swim 10 yards or more underwater without taking a breath The third requirement is that the person swims 10 yards or more underwater without taking a breath. The phrase or more is another way to say greater than or equal to. Thus, we can use the following inequality to represent this requirement. u≥10 To graph this inequality, we will place a closed circle at 10, because u can equal 10, and shade the region to the right on the number line.bFor this exercise, we must determine if given values lie within the solution set for each inequality we created. We will first translate the given criteria into equations. Then we will plot a point on the corresponding graph for each equation.Swim at least 100 yards It is given that you can swim 250 feet. We must first convert this into yards so we can compare the values. There are 3 feet in 1 yard, so we can multiply 250 feet by the conversion factor 3feet1yard​ to convert 250 feet to yards. 250 feet⋅3 feet1 yard​ Simplify expression a⋅cb​=ca⋅b​3 feet250 feet⋅1 yard​Cross out common factors3 feet250 feet⋅1 yard​Simplify quotient3250⋅1 yard​a⋅1=a3250 yards​Calculate quotient83.33333… yards 83.33 yards Thus, we can swim 83.33 yards. This means s=83.33. We will place a point on the number line graph that we made for swimming at least 100 yards.Notice that the point for s=83.33 does not lie within the shaded region on the graph. Thus, you do not meet this requirement.Tread water for at least 5 minutes It is given that you can tread water for 6 minutes. Thus, t=6. We will place a point on the number line graph made for treading water.Notice that the point for t=6 lies within the shaded region on the graph. Thus, you meet this requirement.Swim 10 yards or more underwater without taking a breath It is given that you can swim 35 feet underwater without taking a breath. We must first convert this into yards so we can compare the values. There are 3 feet in 1 yard, so we can mutliply 35 feet by the conversion factor 3 feet1 yard​ to convert 35 feet to yards. 35 feet⋅3 feet1 yard​ Simplify expression a⋅cb​=ca⋅b​3 feet35 feet⋅1 yard​Cross out common factors3 feet35 feet⋅1 yard​Simplify quotient335⋅1 yard​a⋅1=a335 yards​Calculate quotient11.66666… yards 11.66 yards Thus, we can swim 11.67 yards underwater without taking a breath. This means u=11.67. We will place a point on the number line graph made for swimming underwater.Notice that the point for u=11.67 lies within the shaded region on the graph. Thus, you meet this requirement.Conclusion You meet two of the three requirements for the course. Thus, you do not satisfy all of the requirements.
Exercises 15 If the maximum volume of a pelican's bill is 700 cubic inches, the total volume inside of pelican's bill must be less than or equal to that. We can express this algebraically as: …≤700. The pelican has already scooped 100 cubic inches of water into its bill, so the sum of that and any additional volume must be less than or equal to 700. If we let v represent the additional volume the pelican can scoop, our inequality becomes: 100+v≤700. Solving this inequality for v will tell us how much more water the pelican can scoop. 100+v≤700 LHS−100≤RHS−100 LHS−100≤RHS−100100+v−100≤700−100Subtract terms v≤600 The pelican's bill can contain up to 600 additional cubic inches of water.
Exercises 16 aIt is given that the cost of a bike is at least 120 dollars. This means the price can be greater than or equal to 120. It is also given that you save 15 dollars per week. If we let w be the number of weeks you've saved, we can write the following inequality. 15w≥120 To solve the inequality, we must isolate the variable. 15w≥120LHS/15≥RHS/15w≥8 The solution w≥8 means that it will take you at least 8 weeks to save enough money to afford a bike.bIt is given that your parents contribute 65 dollars to your savings. This changes the inequality as it adds to the 15 dollars per week you were saving. Thus, we can now write the inequality 15w+65≥120 to represent the number of weeks it will take you to save. Since your parents gave you some money, it will take a shorter amount of time to save up the necessary amount. Solving the new inequality will tell us how long. 15w+65≥120LHS−65≥RHS−6515w≥55LHS/15≥RHS/15w≥3.67 The solution w≥3.67 means that it will take you at least 3.67 weeks to save enough money to buy a bike if your parents contribute 65 dollars.