QA #3 Equations, Expressions and Inequalities

Questions and Answers Sheet 3 Equations, Expressions and Inequalities Question #1 (a) What is the logical first step in solving the equation |2𝑥 − 1| = 5? (b) What is the logical first step in solving the inequality |3𝑥 + 2| ≤ 8? Answer: Step 1 (a) Consider the given equation |2𝑥 − 1| = 5 . In this equation, only absolute value term is on one side of the equation. Hence, we can use the property of absolute value to obtain two equations. Thus, |2𝑥 − 1| = 5 is equivalent to 2𝑥 − 1 = 5 𝑎𝑛𝑑 2𝑥 − 1 = −5 . That is, the logical step in solving the equation |2𝑥 − 1| = 5 will be to use the property of absolute value which says |𝑥 | = 𝑐 is equivalent to |𝑥 | = ±𝑐 . Hence, the given equation |2𝑥 − 1| = 5 is equivalent to 2𝑥 − 1 = 5 𝑎𝑛𝑑 2𝑥 − 1 = −5 . Step 2 (b) Consider the given inequality |3𝑥 + 2| ≤ 8 . As it is clear from the above table, solution set of inequality can be written as −8 ≤ 3𝑥 + 2 ≤ 8 . Once we have written the inequality in this form, we can split and write it in terms of two inequalities 3𝑥 + 2 ≥ −8 𝑎𝑛𝑑 3𝑥 + 2 ≤ 8 . The first logical step in solving the inequality |3𝑥 + 2| ≤ 8 would be to use the properties of absolute value inequalities and obtain the following inequalities. 3𝑥 + 2 ≥ −8 𝑎𝑛𝑑 3𝑥 + 2 ≤ 8 . Question #2 Min 𝑃 = 4𝑥 + 10𝑦 Subject to: 2𝑥 + 6𝑦 ≥ 24 4𝑥 + 2𝑦 ≥ 18 Answer: Step 1 Let's solve the two inequalities treating them as equations and then see if the solution of the system of equations violate any of the inequalities. 2𝑥 + 6𝑦 = 24 ----------------Eqn (1) 4𝑥 + 2𝑦 = 18 ---------------Eqn (2) Eqn (1) − 3 eqn (2) gives: (2𝑥 − 12𝑥 ) = 24 − 3 × 18 Or, −10𝑥 = −30 Hence, 𝑥 = 3 From eqn (1): 𝑦 = (24 − 2 × 3)/6 = 3 Step 2 Hence, the minimum 𝑃 = 4𝑥 + 10𝑦 = 4 × 3 + 10 × 3 = 42 Step 3 Hence, the final answer: 𝑥 = 3, 𝑦 = 3, 𝑃 = 42 Question #3 Describe the strategy you would use to solve log 6 𝑥 − log 6 4 + log 6 8 . a. Use the product rule to turn the right side of the equation into a single logarithm. Recognize that the resulting value is equal to x. b. Express the equation in exponential form, set the exponents equal to each other and solve. c. Use the fact that the logs have the same base to add the expressions on the right side of the equation together. Express the results in exponential form, set the exponents equal to each other and solve. d. Use the fact that since both sides of the equations have logarithms with the same base to set the expressions equal to each other and solve. Answer: log 6 𝑥 = log 6 4 + log 6 8 𝑥 = log 6(32) {log 𝑐 log 6 𝑎 + log 𝑐 𝑏 = log 𝑐 (𝑎𝑏)} Product rule. 𝑥 = 32 Option A is correct. Question #4 1) 2) 3) 4) 4𝑚 3 𝑛 3 4𝑚 −3 𝑛 2 3𝑚𝑛 4 4𝑚 0 3𝑢 −2 3𝑢 −1 𝑣 0 −𝑎 −6 𝑏−5 −2𝑏2 ⋅ (𝑎 3 𝑏−3 )6 Answer: In algebra, to use numerals, symbols, and letters called variables, or pro-numerals, and combinations of both. They stand for the unknown values. Algebraic expressions or equations, the unknown values are represented by pro-numerals or variables. The variable is a symbol that stands for a number. A symbol or letter representing an unknown member of a set. In algebraic expressions, a variable stands for a value. Sometimes it is called an unknown. The same variable may have different values under different conditions. ''Since you have posted a question with multiple sub-parts, we will solve the first four sub-parts for you. To get the remaining sub-part solved, please repost the complete question and mention the sub-parts to be solved.'' 1) Cancel the common factor 4 and simplifying 𝑚3 /𝑚−3 𝑔𝑖𝑣𝑒𝑠 𝑚6 and 𝑛3 /𝑛2 is n using the rule 𝑎/𝑏 = 𝑎𝑏 −1 4𝑚 3 𝑛 3 4𝑚 −3 𝑛 2 𝑚3𝑛3 = 𝑚−3𝑛2 = 𝑚3−(−3) ⋅ 𝑛3−2 = 𝑚6 𝑛 2) Apply the rule 𝑎0 = 1 where 𝑎 ≠ 0 3𝑚𝑛 4 4𝑚 0 = 3𝑚𝑛 4 4⋅1 = 3𝑚𝑛 4 4 3) Apply the exponent rule 𝑎0 = 1 where 𝑎 ≠ 0 ; cancel out the common factor 3 and 𝑢−2 /𝑢−1 is 1/𝑢. 3𝑢 −2 3𝑢 −2 = 3𝑢−1⋅1 3𝑢 −1 𝑣 0 3𝑢 −2 = 3𝑢−1 𝑢 −2 𝑢 −1 = 𝑢−2−(−1) = 𝑢−2+1 = 𝑢−1 3𝑢 −2 1 =𝑢 3𝑢 −1 𝑣 0 4) Apply the fraction rule −𝑎/−𝑏 = 𝑎/𝑏 −𝑎 −6 𝑏 −5 −2𝑏 2 ⋅(𝑎 3 𝑏−3 )6 = 𝑎 −6 𝑏−5 = 2𝑏2(𝑎3𝑏−3)6 𝑎 −6 2𝑏2+5 (𝑎 3 𝑏−3 )6 𝑎 −6 = 2𝑏7 (𝑎3𝑏−3)6 𝑎 −6 = 2𝑏7 (𝑎6.3𝑏−18) = = = 𝑎 −6 1 2⋅ 18 𝑎 18 𝑏7 𝑏 1 1 2⋅ 18 𝑎 24 𝑏7 𝑏 1 2𝑎24 𝑏11 𝑏11 = 2𝑎24 Question #5 Simplify the following expressions using the imaginary number i: 1. 5√−12 2. √−8 3. 3√−7 4. −3√−200 Write the following numbers using the imaginary number i, and then perform the operations necessary and simplify your answer. 1. 3√−9 ⋅ 2√−49 2. √−12 ⋅ √−36 3. 𝑖√−36 − 𝑖2 Simplify the following expressions using the imaginary number i when needed: 8. 𝑖8 9. 𝑖75 10. (𝑖2)(𝑖6) Simplify and rationalize the denominators. 11. 4√−3/2 12. −3√−2/27 13. √−4/27 Perform the operation indicated in each problem below: 14. (8 + 6𝑖 ) + (2 + 𝑖 ) 15. −9 + (4 − 2𝑖 ) 16. (1 + 𝑖 ) − (1 − 𝑖 ) 17. 3𝑖 − (4 + 15𝑖 ) Perform the operation indicated in each problem below: 18. (√2 + 3𝑖)(√2 − 3𝑖) 19. (√27 − 3𝑖)(√3 + 𝑖) 20. (√2 − 𝑖√3)(√2 + 𝑖√3) 21. 2𝑖(3𝑖 − 2) 22. (3 − 6𝑖 )2 Determine the sum or difference in geometric terms, and then check it algebraically. Remember that 𝑎 − 𝑏 = 𝑎 + (𝑏 ) . 23. (8 − 5𝑖 ) − (2 + 3𝑖 ) 24. (−6 + 𝑖 ) − (−2 − 3𝑖 ) 25. (3 + 𝑖 ) + (−2 + 2𝑖 ) + (8 − 2𝑖 ) Solve the following equations using the quadratic formula. 26. 𝑋 2 − 4𝑋 + 13 = 0 27. 𝑋 2 + 49 = 0 28. 𝑋 2 + 5𝑋 + 8 = 0 29. 𝑋2 − 3𝑋 + 4 = 0 Answer: 1) The given expression is 5√−12 Simplify the expression 5√−12 = 5√−1√12 = 5𝑖√4√3 = 5𝑖(2)√3 = 10𝑖√3 2) The given expression is √−8 Simplify the expression √−8 = √−1√8 = 𝑖 √4√2 = 2𝑖√2 3) The given expression is 3√−7 Simplify the expression 3√−7 = 3√−1√7 = 3𝑖√7 4) The given expression is −3√−200 . Simplify the expression −3√−200 = −3√−1√200 = −3𝑖 √2√100 = −3𝑖(10)√2 = −30𝑖√2 Question #6 Find the standard equation of any parabola that has vertex 𝑉. 𝑉 (−3,1) Answer: Step 1 The equation is a statement that consists of equal symbol between two algebraic expressions. The solution for the variable of the equation must satisfy the equation when we resubstitute the solution in the equation. Step 2 Quadratic equations are the polynomial equations of degree 2 in one variable of type 𝑓 (𝑥 ) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 where 𝑎, 𝑏, 𝑐, are real values and 𝑎 ≠ 0 . The equation of a parabola is a quadratic equation. Let it be as 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐. To get the standard form of parabola equation with vertex at4 substitute this point in the vertex equation of parabola and simplify it to write it in standard form as follows; The parabola equation with vertex at (ℎ, 𝑘) is 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 . So, substitute the given vertex point (−3,1) in the vertex equation of parabola with (ℎ, 𝑘) = (−3,1) as follows; 𝑦 = 𝑎 (𝑥 − ℎ )2 + 𝑘 2 𝑦 = 𝑎(𝑥 − (−3)) + 1 ............. Substitute the vertex (ℎ, 𝑘) = (−3,1) . 𝑦 = 𝑎 (𝑥 + 3 )2 + 1 = 𝑎(𝑥 2 + 6𝑥 + 9) + 1 = 𝑎𝑥 2 + 6𝑎𝑥 + 9𝑎 + 1 Hence, the standard equation of the parabola with the vertex at (−3,1) is 𝑦 = 𝑎𝑥 2 + 6𝑎𝑥 + 9𝑎 + 1 , where a is non-zero real number. Question #7 Explain why the solutions of the simultaneous equations 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 𝑑𝑥 + 𝑒𝑦 + 𝑓 = 0 have rational expression in {𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓} whenever 𝑎𝑒 ≠ 𝑏𝑑 . Answer: Step 1 given simultaneous equations are, 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 and 𝑑𝑥 + 𝑒𝑦 + 𝑓 = 0 Step 2 given system can be written as, 𝑎𝑥 + 𝑏𝑦 = −𝑐 and 𝑑𝑥 + 𝑒𝑦 ± 𝑓 above system is non homogeneous system of equation, compaire it with 𝑎𝑥 = 𝑏  a b   x   −c  d e   y  = − f       when 𝑎𝑒 = 𝑏𝑑 then we get a solution in integer. and if 𝑎𝑒 ≠ 𝑏𝑑 then we have to solve system of simultaneous equation by using row transformation.  a b   x   −c  = ( r3 = r3 − 9r2 )  d e   y   − f   1 1 R1    a d b  c  x  −   a  = a  y   e    − f  𝑅2 = 𝑅2 − 𝑑𝑅1 gives, b   c   1 a   x  −    = a   bd y 0 e − ( )    − f + cd     a  by solving x and y we get rational expressions, so we have a rational expressions in {𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓} whenever 𝑎𝑒 ≠ 𝑏𝑑 . Question #8 Solve the following quadratic expressions by factoring. First write the expressions in completely factored form. Then write the real numer solutions. [Hint: Remember to use proper notation when writing the real number solutions.For example: if the solutions are 𝑥 = 1 and 𝑥 = 4 , write 𝑥 = 1,4 . 1) 𝑥 2 + 9𝑥 + 20 = 0 Factored form - ? Real number solutions - ? 2) 𝑥 2 − 𝑥 − 72 = 0 Factored form - ? Real number solutions - ? 3) 𝑥 2 − 36 = 0 Factored form - ? Real number solutions - ? 4) 𝑥 2 − 10𝑥 + 25 = 0 Factored form - ? Real number solutions - ? Answer: 1) Given: 𝑥 2 + 9𝑥 + 20 = 0 ⇒ 𝑥 2 + 4𝑥 + 5𝑥 + 20 = 0 ⇒ 𝑥 (𝑥 + 4 ) + 5 (𝑥 + 4 ) = 0 ⇒ (𝑥 + 4)(𝑥 + 5) = 0 Factored form (𝑥 + 4)(𝑥 + 5) Real number solution ⇒ (𝑥 + 4)(𝑥 + 5) = 0 ⇒ 𝑥 = −4, 𝑥 = −5 𝑥 = −4, −5 2) ⇒ 𝑥 2 − 𝑥 − 72 = 0 ⇒ 𝑥 2 − 9𝑥 + 8𝑥 − 72 = 0 ⇒ 𝑥 (𝑥 − 9 ) + 8 (𝑥 − 9 ) = 0 ⇒ (𝑥 − 9)(𝑥 + 8) Factored form (𝑥 − 9)(𝑥 + 8) Real number solution ⇒ (𝑥 − 9)(𝑥 + 8) = 0 ⇒ 𝑥 = 9, 𝑥 = −8 𝑥 = 9, −8 3) 𝑥 2 − 36 = 0 ⇒ 𝑥 2 = 36 ⇒ 𝑥 = √36 ⇒𝑥=6 Factored form. 𝑥=6 Real number solution. 𝑥 = 0.6 4) 𝑥 2 − 10𝑥 + 25 = 0 ⇒ 𝑥 2 − 5𝑥 − 5𝑥 + 25 = 0 ⇒ 𝑥 2 − 5𝑥 − 5𝑥 + 25 = 0 ⇒ 𝑥 (𝑥 − 5 ) − 5 (𝑥 − 5 ) = 0 ⇒ (𝑥 − 5)(𝑥 − 5) = 0 Factored form (𝑥 − 5)(𝑥 − 5) Real number solution (𝑥 − 5)(𝑥 − 5) = 0 𝑥 = 5,5 Question #9 Write 𝑎, 𝑏, 𝑐, and 𝑑 in least to greatest. Given the inequality equation: 1 5 𝑏 > 𝑑 + 3 ; 𝑐 + 1 < 𝑎 − 4; 𝑑 + 8 > 𝑎 + 2 Answer: Concept used: In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expression. It is used most often to compare two numbers on the number line by their size. An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value. 𝑎 ≠ 𝑏 says that a is not equal to b. 𝑎 < 𝑏 says that 𝑎 is less than 𝑏. 𝑎 > 𝑏 says that 𝑎 is greater than 𝑏. There are four different types of inequalities: Greater than -(>); Less than -(<); Greater than or equal to −(≥) ; Less than or equal to -(≤) For inequality equation: If 𝑏 > 𝑐 ⇒ 𝑐 < 𝑏 𝑜𝑟 𝑏 < 𝑐 ⇒ 𝑐 > 𝑏 Rules for solving inequality equations: These things do not affect the direction of the inequality: -Add (or subtract) a number from both sides -Multiply (or divide) both sides by a positive number -Simplify a side But these things do change the direction of the inequality (''<''becomes''>'' for example): -Multiply (or divide) both sides by a negative number -Swapping left and right hand sides Calculation: Check each inequality equation: 𝑐+1 <𝑎−4 𝑐+1−1 <𝑎−4−1 𝑐 < 𝑎−5 5 𝑑+8> 𝑎+2 5 5 5 𝑑+8−8>𝑎+2−8 𝑑>𝑎+ 16 5 − 8 8 𝑑>𝑎+ 11 8 1 𝑏 >𝑑+3 Let 𝑎 = 10 ; 𝑐 < 𝑎 − 5 ⇒ 𝑐 < 10 − 5 ⇒ 𝑐 < 5 𝑑>𝑎+ 11 8 ⇒ 𝑑 > 10 + 1.375 ⇒ 𝑑 > 11.375 1 𝑏 > 𝑏 + 3 ⇒ 𝑏 > 11.375 + 0.33 ⇒ 𝑏 > 11.705 if 𝑎 = 10 , 𝑡ℎ𝑒𝑛 𝑏 > 11.705; 𝑐 < 5𝑑 > 11.375 Order from least to greatest: 𝑐 < 𝑎 < 𝑑 < 𝑏 Thus, the order from least to greatest: 𝑐 < 𝑎 < 𝑑 < 𝑏 Question #10 Graphs of the functions 𝑓 and 𝑔 are given. a) Which is larger, 𝑓(0) or 𝑔(0)? b) Which is larger,𝑓 (−3) or 𝑔(−3)? c) For which values of 𝑥 is 𝑓 (𝑥 ) = 𝑔(𝑥 )? Answer: A single variable function can be defined as a relation between the function variable and the function value which varies with the function variable. And if the function has one unique value for each value of x, it is called a one to one function Step 2 a) 𝑓(0)and 𝑔(0) implies the value of the function y when the value of x is zero. From the graph at 𝑥 = 0 , the value of the function 𝑓 (𝑥 ) is 3 and the value of 𝑔(𝑥 ) is around 0.5. Therefore at 𝑥 = 0, 𝑓 (𝑥 ) > 𝑔(𝑥 ) b) As in the above question𝑓 (−3) and 𝑔(−3) implies the value of the function y, when the value of x is -3. From the graph of the functions, the value of f(x) at 𝑥 = −3 is -1 and the value of 𝑔(𝑥 ) at 𝑥 = −3 is 2. Therefore 𝑔(−3) is greater than𝑓(−3) c) The point where 𝑓(𝑥 ) = 𝑔(𝑥 ) is the point where the graphs of the two functions intersects. From the graph, the two functions intersects at the points 𝑥 = −2,2 . Therefore for 𝑥 = −2 and 2, 𝑓(𝑥)will be equal to 𝑔(𝑥 ). Question #11 Use the Gauss-Jordan method to solve the following system of equations. 3𝑥 − 4𝑦 + 4𝑧 = 10 3𝑥 + 5𝑦 − 𝑧 = 15 12𝑥 − 7𝑦 + 11𝑧 = 45 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution is ( _ , _ ,_ ) in the order x, y, z. (Simplify your answers.) B. There is an infinite number of solutions. The solution is ( _ , _, _ ) where z is any real number. (Simplify your answers. Use integers or fractions for any numbers in the expressions.) C. There is no solution. Answer: Step 1 3𝑥 − 4𝑦 + 4𝑧 = 10 3𝑥 + 5𝑦 − 𝑧 = 15 12𝑥 − 7𝑦 + 11𝑧 = 45 Step 2 By using Gauss-Jordan method solve the system of equations as follows. 1 3 −4 4 10 ( 3 5 −1 15) = ( 3 12 −7 11 45 12 1 =(0 12 −4 3 9 −7 −4 4 10 3 3 3 r 5 −1 15)( r1 = 1 3 −7 11 45 4 10 3 3 −5 5 )( r2 = r2 − 3r1 ) 11 45 −4 3 9 9 4 10 3 3 −5 5 )( r3 = r3 − 12r1 ) −5 5 1 −4 3 = (0 1 0 9 4 10 3 3 r −5 5 )( r2 = 2 ) 9 9 9 −5 5 1 = (0 0 Step 3 On further simplification, 16 110 27 27 3 −4 4 10 4r −5 5 ( 3 5 −1 15) = (0 1 )( r1 = r1 + 2 9 9 3 12 −7 11 45 0 9 −5 5 1 0 16 110 27 27 −5 5 = (0 1 ) ( r3 = r3 − 9r2 ) 9 9 0 0 0 0 1 0  a b   x   −c    =   d e   y  − f  Step 4  From the above calculation there are infinitely many solutions. Let z is a real number so, 𝑥 = 110 27 16 5 − 27 𝑧 𝑎𝑛𝑑 𝑦 = 9 (1 + 𝑧) 110 16 5 Therefore, the solution of the given system of equation is ( 27 − 27 𝑧, 9 (1 + 𝑧), 𝑧) . Question #12 Graphs of the functions f and g are given. a) Which is larger, f(6) or g(6)? b) Which is larger, f(3) or g(3)? c) Find the values of x for which 𝑓 (𝑥 ) = 𝑔(𝑥 ) . Answer: a) 𝑓(6) = 3.5 and 𝑔(6) = 5.5 Clearly 𝑔(6) is larger. b) 𝑓(3) = 6 and 𝑔(3) = 3 Clearly 𝑓(3)is larger. c) From the graph we can see that the two curves meets at 𝑥 = 2 𝑍𝑆𝐾 and 𝑃𝑆𝐾 𝑥 = 7 Hence, 𝑓 (𝑥 ) = 𝑔(𝑥 ) when 𝑥 = 2, 7. Question #13 Solve for the unknown variable from the inequality equation. Given: 1 The inequality equation: 𝑃𝑆𝐾 − 2 𝑛 ≤ 6 Answer: Concept used: In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expression. It is used most often to compare two numbers on the number line by their size. An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value. 𝑎 ≠ 𝑏 says that a is not equal to b. 𝑎 < 𝑏 says that a is less than b. 𝑎 > 𝑏 says that a is greater than b. There are four different types of inequalities: Greater than -(>); Less than -(<); Greater than or equal to −(≥) ; Less than or equal to -(≤) For inequality equation: If 𝑏 > 𝑐 ⇒ 𝑐 < 𝑏 𝑜𝑟 𝑏 < 𝑐 ⇒ 𝑐 > 𝑏 Rules for solving inequality equations: These things do not affect the direction of the inequality: -Add (or subtract) a number from both sides -Multiply (or divide) both sides by a positive number -Simplify a side But these things do change the direction of the inequality (''<''becomes''>'' for example): -Multiply (or divide) both sides by a negative number -Swapping left and right hand sides Calculation: 1 The inequality equation: − 2 𝑛 ≤ 6 Solve: 1 − 𝑛≤6 2 1 −2 ⋅ − 𝑛 ≥ −2 ⋅ 6 [Multiply both sides by -2 and reverse the inequality] 2 𝑛 ≥ −12 Solution of the inequality equation is 𝑛 ≥ −12 Solution set: [−12, ∞) A closed, or shaded, circle is used to represent the inequalities greater than or equal to (\geq) or less than or equal to (≤) . The point is part of the solution. Solution set on the number line. 1 Thus, the solution of the inequality equation − 2 𝑛 ≤ 6𝑍𝑆𝐾𝑖𝑠𝑃𝑆𝐾𝑛 ≥ −12 . Question #14 Solve for the unknown variable from the inequality equation. Given: The inequality equation: 𝑏 + 2 ≥ 4 Answer: Concept used: In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expression. It is used most often to compare two numbers on the number line by their size. An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value. 𝑎 ≠ 𝑏 says that a is not equal to b. 𝑎 < 𝑏 says that a is less than b. 𝑎 > 𝑏 says that a is greater than b. There are four different types of inequalities: Greater than -(>); Less than -(<); Greater than or equal to −(≥) ; Less than or equal to −(≤) For inequality equation: If 𝑏 > 𝑐 ⇒ 𝑐 < 𝑏 𝑜𝑟 𝑏 < 𝑐 ⇒ 𝑐 > 𝑏 Rules for solving inequality equations: These things do not affect the direction of the inequality: -Add (or subtract) a number from both sides -Multiply (or divide) both sides by a positive number -Simplify a side But these things do change the direction of the inequality (''<''becomes''>'' for example): -Multiply (or divide) both sides by a negative number -Swapping left and right hand sides Calculation: The inequality equation: 𝑏 + 2 ≥ 4 Solve: 𝑏 + 2 ≥ 4 𝑏 ± 2 ≥ 4 − 2 [Subtract 2 from both sides] 𝑏 ≥2 A closed, or shaded, circle is used to represent the inequalities greater than or equal to (≥) or less than or equal to (≤). The point is part of the solution. Solution set on the number line Solution of the inequality equation is 𝑏 ≥ 2 Solution set: [2, ∞) Thus, the solution of the inequality equation 𝑏 + 2 ≥ 4 𝑖𝑠 𝑏 ≥ 2 . Question #15 Solve for the unknown variable from the inequality equation. Given: The inequality equation: 𝑟 − 8 ≤ 7 Answer: Concept used: In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expression. It is used most often to compare two numbers on the number line by their size. An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value. 𝑎 ≠ 𝑏 says that a is not equal to b. 𝑎 < 𝑏 says that a is less than b. 𝑎 > 𝑏 says that a is greater than b. There are four different types of inequalities: Greater than -(>); Less than -(<); Greater than or equal to −(≥) ; Less than or equal to −(≤) For inequality equation: If 𝑏 > 𝑐 ⇒ 𝑐 < 𝑏 𝑜𝑟 𝑏 < 𝑐 ⇒ 𝑐 > 𝑏 Rules for solving inequality equations: These things do not affect the direction of the inequality: -Add (or subtract) a number from both sides -Multiply (or divide) both sides by a positive number -Simplify a side But these things do change the direction of the inequality (''<''becomes''>'' for example): -Multiply (or divide) both sides by a negative number -Swapping left and right hand sides Calculation: The inequality equation: 𝑟 − 8 ≤ 7 Solve: 𝑟−8 ≤7 𝑟 − 8 + 8 ≤ 7 + 8 [Add 6 to both sides] 𝑟 ≤ 15 A closed, or shaded, circle is used to represent the inequalities greater than or equal to (≥) or less than or equal to (≤) . The point is part of the solution. Solution set on the number line Solution of the inequality equation is 𝑟 ≤ 15 Solution set: (−∞, 15] Thus, the solution of the inequality equation 𝑟 − 8 ≤ 7 𝑖𝑠 𝑟 ≤ 15 . Question #16 Solve for the unknown variable from the inequality equation. Given: The inequality equation: 𝑝 − 6 ≥ 3 Answer: Concept used: In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expression. It is used most often to compare two numbers on the number line by their size. An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value. 𝑎 ≠ 𝑏 says that a is not equal to b. 𝑎 < 𝑏 says that a is less than b. 𝑎 > 𝑏 says that a is greater than b. There are four different types of inequalities: Greater than -(>); Less than -(<); Greater than or equal to −(≥); Less than or equal to −(≤) For inequality equation: If 𝑏 > 𝑐 ⇒ 𝑐 < 𝑏 𝑜𝑟 𝑏 < 𝑐 ⇒ 𝑐 > 𝑏 Rules for solving inequality equations: These things do not affect the direction of the inequality: -Add (or subtract) a number from both sides -Multiply (or divide) both sides by a positive number -Simplify a side But these things do change the direction of the inequality (''<''becomes''>'' for example): -Multiply (or divide) both sides by a negative number -Swapping left and right hand sides Calculation: The inequality equation: 5 Solve: 𝑝−6 ≥3 𝑝 − 6 + 6 ≥ 3 + 6 [Add 6 to both sides] 𝑝 ≥9 A closed, or shaded, circle is used to represent the inequalities greater than or equal to (≥) or less than or equal to (≤). The point is part of the solution. Solution set on the number line Solution of the inequality equation is 𝑝 ≥ 9 Solution set: [9, ∞) Thus, the solution of the inequality equation 𝑝 − 6 ≥ 3 𝑖𝑠 𝑝 ≥ 9 . Question #17 Solve for the unknown variable from the inequality equation. Given: The inequality equation: 5 + 𝑐 ≤ 1 Answer: Concept used: In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expression. It is used most often to compare two numbers on the number line by their size. An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value. 𝑎 ≠ 𝑏 says that a is not equal to b. 𝑎 < 𝑏 says that a is less than b. 𝑎 > 𝑏 says that a is greater than b. There are four different types of inequalities: Greater than -(>); Less than -(<); Greater than or equal to -(\geq) ; Less than or equal to −(≤) For inequality equation: If 𝑏 > 𝑐 ⇒ 𝑐 < 𝑏 𝑟 < 𝑐 ⇒ 𝑐 > 𝑏 Rules for solving inequality equations: These things do not affect the direction of the inequality: -Add (or subtract) a number from both sides -Multiply (or divide) both sides by a positive number -Simplify a side But these things do change the direction of the inequality (''<''becomes''>'' for example): -Multiply (or divide) both sides by a negative number -Swapping left and right hand sides Calculation: The inequality equation: 5 + 𝑐 ≤ 1 Solve: 5+𝑐 ≤1 5 − 5 + 𝑐 ≤ 1 − 5 [Subtract 5 from both sides] 𝑐 ≤ −4 A closed, or shaded, circle is used to represent the inequalities greater than or equal to (\geq) or less than or equal to (≤) . The point is part of the solution. Solution set on the number line Solution of the inequality equation is 𝑐 ≤ −4 Solution set: (−∞, −4] Thus, the solution of the inequality equation 5 + 𝑐 ≤ 1 𝑖𝑠 𝑐 ≤ −4 . Question #18 Solve the following quadratic expressions by factoring. First write the expressions in completely factored form. Then write the real numer solutions. [Hint: Remember to use proper notation when writing the real number solutions.For example: if the solutions are 𝑥 = 1 and 𝑥 = 4 , write 𝑥 = 1,4 . 1) 𝑥 2 − 5𝑥 = 0 Factored form - ? Real number solutions - ? 2) 4𝑥 2 + 8𝑥 = 0 Factored form - ? Real number solutions - ? 3) −7𝑥 2 − 21𝑥 = 0 Factored form - ? Real number solutions - ? 4) 4𝑥 2 − 8𝑥 = 0 Factored form - ? Real number solutions - ? Answer: 1) 𝑥 2 − 5𝑥 = 0 𝑥 (𝑥 − 5 ) = 0 𝑥 = 0 𝑎𝑛𝑑 𝑥 = 5 Factored form = 𝑥(𝑥 − 5) real number solution = 0,5 2) 4𝑥 2 + 8𝑥 = 0 4𝑥(𝑥 + 2) = 0 𝑥 = 0 𝑎𝑛𝑑 𝑥 = −2 Factored form = 4𝑥(𝑥 + 2) real number solution = 0, −2 3) −7𝑥 2 − 21𝑥 = 0 −7𝑥(𝑥 + 3) = 0 𝑥 = 0 𝑎𝑛𝑑 𝑥 = −3 Factored form = −7𝑥(𝑥 + 3) real number solution = 0, −3 4) 4𝑥 2 − 8𝑥 = 0 4𝑥(𝑥 − 2) = 0 𝑥 = 0 𝑎𝑛𝑑 𝑥 = 2 Factored form = 4𝑥(𝑥 − 2) real number solution = 0,2 Question #19 The reduced row-echelon form of the augmented matrix for a system of linear equations with variables 𝑥1 , ⋯ , 𝑥4 is given below. Determine the solutions for the system and enter them below. 1 0 −1 −5 | 0  0 1 3 −3 | −1 If the system has infinitely many solutions, select ''The system has at least one   solution''. Your answer may use expressions involving the parameters r, s, and t. The system has at least one solution 𝑥1 = 0 𝑥2 = 0 𝑥3 = 0 𝑥4 = 0 Answer: Step 1 Since rank(augmented matrix) 0 and 𝑥 ≻ 1 Answer: Step 1 The given compound inequality is, 𝑥 > 0 ⋯ ⋯ (𝑖 ) 𝑥 ≻ 1 ⋯ ⋯ (𝑖 ) To find the solution of the compound inequality, we find the common set of solutions which satisfies both the inequality. For the first inequality, the value of 𝑥 should be greater than 0 and for the second inequality, the value of 𝑥 should be −1. If we take 𝑎 value of 𝑥 as greater 0. then it will satisfy both the equations. Step 2 Hence, the graph of the two inequalities are as, Step 3 Therefore, the required solution set is as, 𝑥 ∈ (0, ∞)