Absolute Value. Example. Practice
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2 Absolute Value The absolute value of a number is the distance between the origin of a number line and the point representing that number. Look at the number line below. Both 7 and 7 are 7 units from the origin. 7 units 7 units The notation for absolute value is a and is read as the absolute value of a. Rules for the Absolute Value of a Number 1. If a is a positive number, then a a (e.g. 4 4).. If a is zero, then a 0 (e.g. 0 0). 3. If a is a negative number, then a a (e.g. 4 4). Solve the equation. x 15 Step 1 Which positive number is 15 units from the origin? Step Which negative number is 15 units from the origin? Step 3 Check that both solutions are true true true Solve. 1. x 10 Which positive number is 10 units from the origin? Which negative number is 10 units from the origin? Check that both solutions are true. 10. x 3 5. x x x 4. 7 x 7. 6 x Saddleback Educational Publishing 006 (888)
3 Absolute Value Equations The absolute value of a number is the distance between the origin of a number line and the point representing that number. To solve an absolute value equation you need to account for the value inside the absolute value symbol being positive, and the value inside the absolute value symbol being negative. Rules for Solving an Absolute Value Equation 1. Account for the value of the expression inside the absolute value symbol being positive.. Account for the value of the expression inside the absolute value symbol being negative. Solve. x + 10 Step 1 Account for the value of the expression inside the absolute value symbol being positive. Step Account for the value of the expression inside the absolute value symbol being negative. Solve. 1. x 1 9 Account for the value of the expression inside the absolute value symbol being positive. Account for the value of the expression inside the absolute value symbol being negative. What value of x will result in an amount of 10? 8 x True What value of x will result in an answer of 10? 1 x True The solution is 1 and 8. What value of x will result in an answer of 9? x What value of x will result in a value of 9? x x x 1 3. x x Saddleback Educational Publishing 006 (888)
4 Compound Inequalities A compound inequality is made of two inequalities connected by and or or. A compound inequality with and : 4 < x < 10 A compound inequality with or : x < or x > 5 Rules for Solving a Compound Inequality Compound inequality with and (x is in the middle of the expression). 1. Write the original inequality as two inequalities.. Solve for the left side of the inequality. 3. Solve for the right side of the inequality. Compound inequality with or. 1. Write the original inequality.. Solve the left side of the inequality. 3. Solve the right side of the inequality. Solve. 4 x < 10 (Compound inequality with and. ) Step 1 Write the original inequality as two inequalities. Step Solve for the left side of the inequality. Step 3 Solve for the right side of the inequality. 4 x < 10 4 x and x < 10 4 x x x < 10 x < 5 x < 5 Solve. 1. 3x < 6 or x + > 10 (Compound inequality with or. ) Write the original inequality. 3x < 6 or x + > 10 Solve the left side of the inequality. Solve the right side of the inequality.. 6x + < 10 or 4x > x + 3 > 1 or 4x > 4 3x < 6 x + > 10 The solution is. 4 Saddleback Educational Publishing 006 (888)
5 Absolute Value Inequalities When solving an absolute inequality you can apply what you know from solving absolute value equations. As with absolute value equations, you look for values that are positive and negative that make the inequality true. Rules for Solving Absolute Value Inequalities 1. Write the absolute value inequality as an inequality with the original number to the right of the inequality symbol.. Write the absolute value inequality as an inequality with the inequality symbol reversed and the inverse of the number to the right of the symbol. 3. Solve each inequality. Write the solutions as a compound inequality. Solve. x + > 10 Step 1 Write the absolute value inequality as an inequality with the original number to the right of the inequality symbol. Step Write the absolute value inequality as an inequality with the inequality symbol reversed and the inverse of the number to the right of the symbol. Step 3 Solve each inequality. Write the solutions as a compound inequality. x + > 10 x + > 10 x + > 10 x + < 10 x + > 10 x > 8 x + < 10 x < 1 x < 1 or x > 8 Solve. 1. x + 4 < 1 Write the absolute value inequality as an inequality with the original number to the right of the inequality symbol. Write the absolute value inequality as an inequality with the inequality symbol reversed and the inverse of the number to the right of the symbol. Solve each inequality. Write the solutions as a compound inequality. x + 4 < 1 x + 4 < 1 x < x > The solution is.. x x 15 > 5 3. x x > 5 Saddleback Educational Publishing 006 (888)
6 Graphing Absolute Value Inequalities The absolute value of a number is the distance between the origin of a number line and the point representing that number. When graphing an absolute value inequality, you first have to solve the inequality, treating it as a compound inequality. Then graph each solution. Rules for Graphing an Absolute Value Inequality 1. Rewrite the inequality as two inequalities.. Solve each inequality. 3. Graph each solution. If the solution is connected by or, then the graph is away from the two points. If the solution is connected by and, then the graph is between the two points. Solve, then graph the solution. x + 6 < 11 Step 1 Rewrite the inequality as two inequalities. Step Solve each inequality. Step 3 Graph each solution. If the solution is connected by or, then the graph is away from the two points. If the solution is connected by and, then the graph is between the two points. x + 6 < 11 and x + 6 > 11 x < 5 and x > Solve, then graph the solution. 1. x Rewrite the inequality as two inequalities. x or Solve each inequality. x or x Graph each solution. If the solution is connected by or, then the graph is away from the two points. If the solution is connected by and, then the graph is between the two points.. x + 4 > 5 3. x Saddleback Educational Publishing 006 (888)
7 Introduction to Matrices A matrix is a rectangular array of numbers written within brackets. A matrix is identified by a capital letter. A matrix is classified by its dimensions the number of columns and rows it contains. 9,300,900 Matrix X to the right has 3 rows and columns. It is a X 3 matrix. 3,00,100 A matrix element is a number in the matrix. 15,400 1,00 Each matrix element is identified by its location within the matrix. columns Rules for Reading a Matrix 1. The dimensions of a matrix are given in terms of rows and columns.. A matrix element is identified by (1) using the letter of the matrix, and () using a subscript to identify the position of the element by row and column. State the dimensions of the matrix. Identify element A 3. A Step 1 The dimensions of a matrix are given in terms of rows and columns. Step A matrix element is identified by (1) using the letter of the matrix, and () using a subscript to identify the position of the element by the row and column The matrix has rows and 3 columns; it is a 3 matrix. A 3 is the element in row, column 3. A 3 element X 1 3 rows State the dimensions of the matrix. Identify the specified element. 1. Identify element B. The dimensions of a matrix are given in terms of rows and columns. A matrix element is identified by (1) using the letter of the matrix, and () using a subscript to identify the position of the element by the row and column.. Identify element Z 1. Z B The matrix has rows and columns; it is a matrix. B is the element in row, column. B Identify the location of 10. Z Saddleback Educational Publishing 006 (888)
8 Matrix Addition When adding matrices, you add the corresponding elements in each matrix. corresponding elements Rule for Matrix Addition Add corresponding elements in each matrix to form one large matrix. Add Add corresponding elements in each matrix to form one large matrix ( 9) ( 10) ( 3) Add Add corresponding elements in each matrix to form one large matrix Saddleback Educational Publishing 006 (888)
9 Matrix Subtraction When subtracting matrices, you subtract the corresponding elements in each matrix. corresponding elements Rule for Matrix Subtraction Subtract corresponding elements in each matrix to form one large matrix. Subtract Subtract corresponding elements in each matrix to form one large matrix ( 4) Subtract Subtract corresponding elements in each matrix to form one large matrix Saddleback Educational Publishing 006 (888)
10 Scalar Multiplication A matrix is a rectangular arrangement of numbers in rows and columns. You can think of a matrix as a way to organize data, similar to the way data is displayed in a table. A scalar is a real number factor by which all the elements of a matrix are multiplied. Rule for Scalar Multiplication Create an expanded matrix by multiplying each element by the scalar. Solve Create an expanded matrix by 6 4 multiplying each element by the scalar Solve Create an expanded matrix by multiplying each element by the scalar Saddleback Educational Publishing 006 (888)
11 Matrix Multiplication When multiplying matrices, you multiply the elements of a row in the first matrix by the corresponding elements in a column of the second matrix. You then add the products Rules for Matrix Multiplication 1. Circle each row of the first matrix; circle each column of the second matrix.. Multiply the elements of a row in the first matrix by the elements of each column in the second matrix. Add the products in each row. The dimensions of the resulting matrix will be the number of rows in the first matrix by the number of columns in the second matrix. Multiply Step 1 Identify the elements to be multiplied. Step Multiply the elements of a row in the first matrix by the elements of each column in the second matrix. Add the products in each row Multiply Identify the elements to be multiplied Multiply the elements of a row in the first matrix by the elements of each column in the second matrix. Add the products in each row Saddleback Educational Publishing 006 (888)
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