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CALCULUS 1 – Algebra review Intervals and Interval Notation CALCULUS 1 – Algebra review Intervals and Interval Notation Intervals are sets of real numbers. The notation uses square and round brackets to show these sets of numbers. CALCULUS 1 – Algebra review Intervals and Interval Notation Intervals are sets of real numbers. The notation uses square and round brackets to show these sets of numbers. Round bracket – go up to but do not include this number in the set CALCULUS 1 – Algebra review Intervals and Interval Notation Intervals are sets of real numbers. The notation uses square and round brackets to show these sets of numbers. Round bracket – go up to but do not include this number in the set ( 3 , 7 ) - this interval would include all numbers between 3 and 7, but NOT 3 or 7. CALCULUS 1 – Algebra review Intervals and Interval Notation Intervals are sets of real numbers. The notation uses square and round brackets to show these sets of numbers. Round bracket – go up to but do not include this number in the set ( 3 , 7 ) - this interval would include all numbers between 3 and 7, but NOT 3 or 7. Square bracket – include this number in the set CALCULUS 1 – Algebra review Intervals and Interval Notation Intervals are sets of real numbers. The notation uses square and round brackets to show these sets of numbers. Round bracket – go up to but do not include this number in the set ( 3 , 7 ) - this interval would include all numbers between 3 and 7, but NOT 3 or 7. Square bracket – include this number in the set [ 3 , 7 ] - this interval would include all numbers from 3 to 7.. CALCULUS 1 – Algebra review Intervals and Interval Notation When working with equations containing an inequality, the symbols for the inequality determine how you graph and represent the solution as an interval. Round bracket - less than ( < ) , greater than ( > ) CALCULUS 1 – Algebra review Intervals and Interval Notation When working with equations containing an inequality, the symbols for the inequality determine how you graph and represent the solution as an interval. Round bracket - less than ( < ) , greater than ( > ) - open circle on a graph CALCULUS 1 – Algebra review Intervals and Interval Notation When working with equations containing an inequality, the symbols for the inequality determine how you graph and represent the solution as an interval. Round bracket - less than ( < ) , greater than ( > ) - open circle on a graph Square bracket – less than or equal to ( ≤ ), greater than or equal to ( ≥ ) CALCULUS 1 – Algebra review Intervals and Interval Notation When working with equations containing an inequality, the symbols for the inequality determine how you graph and represent the solution as an interval. Round bracket - less than ( < ) , greater than ( > ) - open circle on a graph Square bracket - less than or equal to ( ≤ ), greater than or equal to ( ≥ ) - closed circle on a graph CALCULUS 1 – Algebra review Intervals and Interval Notation Round bracket - less than ( < ) , greater than ( > ) - open circle on a graph Square bracket - less than or equal to ( ≤ ), greater than or equal to ( ≥ ) - closed circle on a graph EXAMPLE : Solve and graph 3x 5 7 and show your answer as an interval CALCULUS 1 – Algebra review Intervals and Interval Notation Round bracket - less than ( < ) , greater than ( > ) - open circle on a graph Square bracket - less than or equal to ( ≤ ), greater than or equal to ( ≥ ) - closed circle on a graph EXAMPLE : Solve and graph 3x 5 7 5 5 3x 12 x4 3x 5 7 and show your answer as an interval CALCULUS 1 – Algebra review Intervals and Interval Notation Round bracket - less than ( < ) , greater than ( > ) - open circle on a graph Square bracket - less than or equal to ( ≤ ), greater than or equal to ( ≥ ) - closed circle on a graph EXAMPLE : Solve and graph 3x 5 7 5 5 3x 12 x4 3x 5 7 and show your answer as an interval 4 graph CALCULUS 1 – Algebra review Intervals and Interval Notation Round bracket - less than ( < ) , greater than ( > ) - open circle on a graph Square bracket - less than or equal to ( ≤ ), greater than or equal to ( ≥ ) - closed circle on a graph EXAMPLE : Solve and graph 3x 5 7 5 5 3x 12 x4 3x 5 7 and show your answer as an interval 4 graph ( 4, ) interval CALCULUS 1 – Algebra review Intervals and Interval Notation Round bracket - less than ( < ) , greater than ( > ) - open circle on a graph Square bracket - less than or equal to ( ≤ ), greater than or equal to ( ≥ ) - closed circle on a graph EXAMPLE # 2 : Solve and graph as an interval 3 2 x 1 5 and show your answer CALCULUS 1 – Algebra review Intervals and Interval Notation Round bracket - less than ( < ) , greater than ( > ) - open circle on a graph Square bracket - less than or equal to ( ≤ ), greater than or equal to ( ≥ ) - closed circle on a graph EXAMPLE # 2 : Solve and graph 3 2 x 1 5 as an interval 3 2x 1 5 1 1 1 2 2x 6 1 x 3 and show your answer CALCULUS 1 – Algebra review Intervals and Interval Notation Round bracket - less than ( < ) , greater than ( > ) - open circle on a graph Square bracket - less than or equal to ( ≤ ), greater than or equal to ( ≥ ) - closed circle on a graph EXAMPLE # 2 : Solve and graph 3 2 x 1 5 and show your answer as an interval 3 2x 1 5 1 1 1 2 2x 6 1 x 3 -1 3 This results in two graphs… x<3 x ≥ -1 CALCULUS 1 – Algebra review Intervals and Interval Notation Round bracket - less than ( < ) , greater than ( > ) - open circle on a graph Square bracket - less than or equal to ( ≤ ), greater than or equal to ( ≥ ) - closed circle on a graph EXAMPLE # 2 : Solve and graph 3 2 x 1 5 and show your answer as an interval 3 2x 1 5 1 1 1 2 2x 6 1 x 3 -1 3 The solution set is where the two graphs overlap ( share ) CALCULUS 1 – Algebra review Intervals and Interval Notation Round bracket - less than ( < ) , greater than ( > ) - open circle on a graph Square bracket - less than or equal to ( ≤ ), greater than or equal to ( ≥ ) - closed circle on a graph EXAMPLE # 2 : Solve and graph 3 2 x 1 5 and show your answer as an interval 3 2x 1 5 1 1 1 2 2x 6 1 x 3 -1 3 The solution set is where the two graphs overlap ( share ) [ -1 , 3 ) interval CALCULUS 1 – Algebra review Intervals and Interval Notation Round bracket - less than ( < ) , greater than ( > ) - open circle on a graph Square bracket - less than or equal to ( ≤ ), greater than or equal to ( ≥ ) - closed circle on a graph EXAMPLE # 3 : Solve and graph as an interval x 2 7 x 12 0 and show your answer CALCULUS 1 – Algebra review Intervals and Interval Notation Round bracket - less than ( < ) , greater than ( > ) - open circle on a graph Square bracket - less than or equal to ( ≤ ), greater than or equal to ( ≥ ) - closed circle on a graph EXAMPLE # 2 : Solve and graph x 2 7 x 12 0 as an interval x 2 7 x 12 0 x 4x 3 0 x 4,3 and show your answer CALCULUS 1 – Algebra review Intervals and Interval Notation Round bracket - less than ( < ) , greater than ( > ) - open circle on a graph Square bracket - less than or equal to ( ≤ ), greater than or equal to ( ≥ ) - closed circle on a graph EXAMPLE # 2 : Solve and graph x 2 7 x 12 0 as an interval x 2 7 x 12 0 x 4x 3 0 x 4,3 These are our critical points and show your answer CALCULUS 1 – Algebra review Intervals and Interval Notation Round bracket - less than ( < ) , greater than ( > ) - open circle on a graph Square bracket - less than or equal to ( ≤ ), greater than or equal to ( ≥ ) - closed circle on a graph EXAMPLE # 2 : Solve and graph x 2 7 x 12 0 and show your answer as an interval x 2 7 x 12 0 x 4x 3 0 x 4,3 These are our critical points Graph the critical points and then use a test point to find “true/false” -4 -3 CALCULUS 1 – Algebra review Intervals and Interval Notation Round bracket - less than ( < ) , greater than ( > ) - open circle on a graph Square bracket - less than or equal to ( ≤ ), greater than or equal to ( ≥ ) - closed circle on a graph EXAMPLE # 2 : Solve and graph x 2 7 x 12 0 and show your answer as an interval x 7 x 12 0 2 x 4x 3 0 x 4,3 These are our critical points Graph the critical points and then use a test point to find “true/false” TRUE FALSE -4 TRUE -3 TEST x = 0 02 70 12 0 0 0 12 0 12 0 TRUE 0 CALCULUS 1 – Algebra review Intervals and Interval Notation Round bracket - less than ( < ) , greater than ( > ) - open circle on a graph Square bracket - less than or equal to ( ≤ ), greater than or equal to ( ≥ ) - closed circle on a graph EXAMPLE # 2 : Solve and graph x 2 7 x 12 0 and show your answer as an interval x 7 x 12 0 2 x 4x 3 0 x 4,3 These are our critical points Graph the critical points and then use a test point to find “true/false” TRUE FALSE -4 TRUE -3 TEST x = 0 02 70 12 0 0 0 12 0 12 0 TRUE 0 ,4 3, interval CALCULUS 1 – Algebra review Absolute Value Equations Remember, absolute value equations have two possible answers; positive and negative. So when solving, drop the absolute value sign, and set the equation equal to the original answer, and also it’s negative counterpart. CALCULUS 1 – Algebra review Absolute Value Equations Remember, absolute value equations have two possible answers; positive and negative. So when solving, drop the absolute value sign, and set the equation equal to the original answer, and also it’s negative counterpart. EXAMPLE # 1 : Solve 2x 5 7 CALCULUS 1 – Algebra review Absolute Value Equations Remember, absolute value equations have two possible answers; positive and negative. So when solving, drop the absolute value sign, and set the equation equal to the original answer, and also it’s negative counterpart. EXAMPLE # 1 : Solve 2x 5 7 2x 5 7 7 2x 5 7 5 5 5 12 2 x 2 12 2 x 2 2 2 2 6 x 1 CALCULUS 1 – Algebra review Absolute Value Equations Remember, absolute value equations have two possible answers; positive and negative. So when solving, drop the absolute value sign, and set the equation equal to the original answer, and also it’s negative counterpart. EXAMPLE # 2 : Solve x2 x2 6 2 CALCULUS 1 – Algebra review Absolute Value Equations Remember, absolute value equations have two possible answers; positive and negative. So when solving, drop the absolute value sign, and set the equation equal to the original answer, and also it’s negative counterpart. EXAMPLE # 2 : Solve x2 x2 6 2 Let u x 2 u2 u 6 u2 u 6 0 u 3u 2 0 Remember u substitution from pre-calc ? CALCULUS 1 – Algebra review Absolute Value Equations Remember, absolute value equations have two possible answers; positive and negative. So when solving, drop the absolute value sign, and set the equation equal to the original answer, and also it’s negative counterpart. EXAMPLE # 2 : Solve x2 x2 6 2 Let u x 2 u2 u 6 u2 u 6 0 u 3u 2 0 Remember u substitution from pre-calc ? x 2 3 0 and x2 2 0 x2 3 and x 2 2 CALCULUS 1 – Algebra review Absolute Value Equations Remember, absolute value equations have two possible answers; positive and negative. So when solving, drop the absolute value sign, and set the equation equal to the original answer, and also it’s negative counterpart. EXAMPLE # 2 : Solve x2 x2 6 2 Let u x 2 u2 u 6 u2 u 6 0 u 3u 2 0 Remember u substitution from pre-calc ? x 2 3 0 and x2 2 0 x2 3 and x 2 2 Can’t have an absolute value equal to a negative answer CALCULUS 1 – Algebra review Absolute Value Equations Remember, absolute value equations have two possible answers; positive and negative. So when solving, drop the absolute value sign, and set the equation equal to the original answer, and also it’s negative counterpart. EXAMPLE # 3 : Solve 2x 3 5 , and show the solution set as an interval. CALCULUS 1 – Algebra review Absolute Value Equations Remember, absolute value equations have two possible answers; positive and negative. So when solving, drop the absolute value sign, and set the equation equal to the original answer, and also it’s negative counterpart. EXAMPLE # 3 : Solve 2x 3 5 5 2 x 3 5 3 3 3 8 2 x 2 8 2x 2 2 2 2 4 x 1 , and show the solution set as an interval. CALCULUS 1 – Algebra review Absolute Value Equations Remember, absolute value equations have two possible answers; positive and negative. So when solving, drop the absolute value sign, and set the equation equal to the original answer, and also it’s negative counterpart. EXAMPLE # 3 : Solve 2x 3 5 5 2 x 3 5 3 3 3 8 2 x 2 8 2x 2 2 2 2 4 x 1 , and show the solution set as an interval. I like to graph the solution to determine the interval… x 1 x4 -1 4 CALCULUS 1 – Algebra review Absolute Value Equations Remember, absolute value equations have two possible answers; positive and negative. So when solving, drop the absolute value sign, and set the equation equal to the original answer, and also it’s negative counterpart. EXAMPLE # 3 : Solve 2x 3 5 5 2 x 3 5 3 3 3 8 2 x 2 8 2x 2 2 2 2 4 x 1 , and show the solution set as an interval. I like to graph the solution to determine the interval… x 1 x4 4 -1 (1,4) interval
