 From here to there and back again  Counts steps  Measures distance

1. Additive inverse - the opposite of a number. Two numbers are additive inverses if their sum is zero 2.Absolute value of x : l x l = 3.Absolute value is non-negative because it is the distance from zero to x. Example for x = l-3l Vocabulary

I am going to my friend’s house. Absolute Value: Counts the Steps

One Absolute Value: Counts the Steps

One, two Absolute Value: Counts the Steps

One, two, three Absolute Value: Counts the Steps

One, two, three, four Absolute Value: Counts the Steps

One, two, three, four, five Absolute Value: Counts the Steps

One, two, three, four, five, six Absolute Value: Counts the Steps

One, two, three, four, five, six, seven Absolute Value: Counts the Steps

One, two, three, four, five, six, seven, eight Absolute Value: Counts the Steps

One, two, three, four, five, six, seven, eight, nine Absolute Value: Counts the Steps 9 steps to go from left to right

It’s time to go home. How many steps to my house? Absolute Value: Counts the Steps

One Absolute Value: Counts the Steps

One, two Absolute Value: Counts the Steps

One, two, three Absolute Value: Counts the Steps

One, two, three, four Absolute Value: Counts the Steps

One, two, three, four, five Absolute Value: Counts the Steps

One, two, three, four, five, six Absolute Value: Counts the Steps

One, two, three, four, five, six, seven Absolute Value: Counts the Steps

One, two, three, four, five, six, seven, eight Absolute Value: Counts the Steps

One, two, three, four, five, six, seven, eight, nine Absolute Value: Counts the Steps 9 steps to go from right to left Do the steps need to be equidistant? Yes

Remember! Absolute value must be nonnegative because it represents _________.

Remember! Absolute value must be nonnegative because it represents distance.

The absolute-value of a number is that number’s distance from zero on a number line. For example, |–9| = ____.

The absolute-value of a number is that numbers distance from zero on a number line. For example, |–9| = 9.

Absolute Value

Represents distance Count the steps

The absolute value of a number is the distance from zero on a number line. Example: |5| read as “the absolute value of 5” Example: |-5| read as “the absolute value of 5” 5 units |5| = 5|–5| = 5 Note: | 5 | = l -5 l= 5

Absolute Value of x l x l the distance from zero to x on a number line

|x| the distance from zero on a number line to x x units 0 x - |x| = x|–x| = x x

Additive inverse The opposite of a number. Two numbers are additive inverses if their sum is zero. x + (-_____) = 0

Additive inverse The opposite of a number. Two numbers are additive inverses if their sum is zero. x + (-x) = 0

Absolute value Is always a positive number except for 0 as 0 is neither positive or negative.

 Graph y = l x l

Absolute-value graphs are V-shaped. axis of symmetry - line that divides the graph into two congruent halves vertex is the “corner" point on the graph.

From the graph of y = |x|, you can tell that: the axis of symmetry is the y-axis (x = 0). the vertex is (0, 0). the domain (x-values) is the set of all real numbers. the range (y-values) is described by y ≥ 0. y = |x| is a function because each domain value has exactly one range value. the x-intercept and the y-intercept are both 0.

Label the axis of symmetry and the vertex. Identify the intercepts, and give the domain and range. Example 1A: Absolute-Value Functions y = |x| + 1 x –2 –

Example 1A Continued the axis of symmetry is the y-axis (x = 0). the vertex is (0, 1). there are no x-intercepts. the y-intercept is +1. the domain is all real numbers. the range is described by y ≥ 1. From the graph you can tell that Axis of symmetry Vertex

Graph Label the axis of symmetry and the vertex. Identify the intercepts, and give the domain and range. Example 1B: Absolute-Value Functions y = |x – 4| x –

the axis of symmetry is x = 4. the vertex is (4, 0). the x-intercept is +4. the y-intercept is +4. the domain is all real numbers. the range is described by y ≥ 0. From the graph you can tell that Example 1B Continued axis of symmetry vertex

Graph Label the axis of symmetry and the vertex. Identify the intercepts, and give the domain and range. f(x) = 3|x| x –2 – Example 1C x

Example 1C Continued the axis of symmetry is x = 0. the vertex is (0, 0). the x-intercept is 0. the y-intercept is 0. the domain is all real numbers. the range is described by y ≥ 0. From the graph you can tell that Axis of symmetry Vertex x

Absolute Value Solve equations in one variable that contain absolute-value expressions.

Represents ________ Absolute Value

Represents distance Absolute Value

Represents distance Counts ____ Absolute Value

Absolute value represents distance Absolute value counts steps Absolute Value

The absolute-value of a number is that number’s distance from zero on a number line. Example |–6| = 6 55 44 33 2 66 11 6 6 units Both 6 and –6 are a distance of 6 units from 0, so both 6 and –6 have an absolute value of 6. 6 units Given |x| = 6 Solve for x This equation asks, “What values of x have an absolute value of 6?” The solutions are 6 and –6. Notice this equation has two solutions.

Example 1A: Solving Absolute-Value Equations Solve for x. |x| = 12 Case 1 x = 12 Case 2 x = –12 The solutions are 12 and –12. Think: Which numbers are 12 units from 0? Rewrite the equation as two cases. Check your work |x| = 12 |12| = = 12 |x| = 12 |  12| = = 12

Example 1B: Solving Absolute-Value Equations 3|x + 7| = 24 |x + 7| = 8 The solutions are 1 and –15. Case 1 x + 7 = 8 Case 2 x + 7 = –8 – 7 = –7 x = 1 x = –15 Since |x + 7| is multiplied by 3, divide both sides by 3 to undo the multiplication. Think: What numbers are 8 units from 0? Rewrite the equations as two cases. Since 7 is added to x subtract 7 from both sides of each equation.

Example 1B Continued 3|x + 7| = 24 The solutions are 1 and –15. Check 3|x + 7| = 24 3|8| = = 24 3|1 + 7| = 24 3(8) = 24 3|  | = = 24 3|  8| = 24 3(8) = 24

Solve each equation. Check your answer. Example 1c |x| – 3 = = +3 |x| = 7 Case 1 x = 7 Case 2 –x = 7 –1(–x) = –1(7) x = –7x = 7 The solutions are 7 and –7. Since 3 is subtracted from |x|, add 3 to both sides. Think: what numbers are 7 units from 0? Rewrite the case 2 equation by multiplying by – 1 to change the minus x to a positive..

Check |x|  3 = 4 7  3 = 4 |7|  3 = 4 4 = 4 |  7|  3 = 4 7  3 = 4 4 = 4 Solve the equation. Check your answer. Example 1c Continued |x|  3 = 4 The solutions are 7 and  7.

Solve the equation. Check your answer. Example 1d |x  2| = 8 Think: what numbers are 8 units from 0? +2 = +2 Case 1 x  2 = 8 x = = +2 x =  6 Case 2 x  2 =  8 Rewrite the equations as two cases. Since 2 is subtracted from x add 2 to both sides of each equation. The solutions are 10 and  6.

Solve the equation. Check your answer. Example 1d Continued |x  2| = 8 The solutions are 10 and  6. Check |x  2| = 8 10  2| = 8 |10  2| = 8 8 = 8 |  6 + (  2)| = = 8 8 = 8

Absolute value equations most often have two solutions, but not all do: 1.If the absolute-value equals 0, there is one solution. 2.If the absolute-value is negative, there are no solutions. Solutions to Absolute Value Equations

Example 2A Solve the equation. Check your answer.  8 = |x + 2|  8 +8 = = |x +2| 0 = x + 2  2 =  2  2 = x Since 8 is subtracted from |x + 2|, add 8 to both sides to undo the subtraction. There is only one case. Since 2 is added to x, subtract 2 from both sides to undo the addition.

Example 2A Continued Solve the equation. Check your answer.  8 = |x +2|  8  8 =  8  8 = |  2 + 2|  8  8 = |0|  8  8 = 0  8 To check your solution, substitute  2 for x in your original equation. Solution is x =  2 Check  8 =|x + 2|  8

Example 2B Solve the equation. Check your answer. 3 + |x + 4| = 0  3 =  3 |x + 4| =  3 Since 3 is added to |x + 4|, subtract 3 from both sides to undo the addition. Absolute values cannot be negative. This equation has no solution.

Remember! Absolute value must be nonnegative because it represents distance.

Example 2c Solve the equation. Check your answer. 2  |2x  5| = 7  2 =  2  |2x  5| = 5 Since 2 is added to  |2x  5|, subtract 2 from both sides to undo the addition. Absolute values cannot be negative. |2x  5| =  5 This equation has no solution. Since |2x  5| is multiplied by a negative 1, divide both sides by negative 1.

Example d Solve the equation. Check your answer.  6 + |x  4| =  6 +6 = +6 |x  4| = 0 x  4 = = +4 x = 4 Since 6 is subtracted from |x  4|, add 6 to both sides to undo the subtraction. There is only one case. Since 4 is subtracted from x, add 4 to both sides to undo the addition.

Solve the equation. Check your answer. Example 2d Continued  6 + |x  4| =  6  6 + |4  4| =  6  6 +|0| =  6  =  6  6 =  6  6 + |x  4| =  6 The solution is x = 4. To check your solution, substitute 4 for x in your original equation.