Before you take a scored quiz for this lesson, try this set of practice questions. How well you score on this self-check will be similar to your scored quiz. If you do not score well on this self-check, please review this lesson and try again.
Which symbol will make this statement true?: length of a pencil ? length of a baseball bat
- >
- <
- ≥
- ≤
A pencil is smaller than a baseball bat. So, the sign should be <
A pencil is smaller than a baseball bat. So, the sign should be <
A pencil is smaller than a baseball bat. So, the sign should be <. > is "greater than or equal to" (also "at least").
A pencil is smaller than a baseball bat. So, the sign should be. <. < is "less than or equal to" (also "no more than").
Bernadette needed to make at least $20 profit from selling the apples from her orchard at $2.00 per carton. Which inequality represents this relationship?
- $2x > $20.00
- $2x < $20.00
- $2x ≥ $20.00
- $2x ≤ $20.00
"at least" means greater than or equal to, >. > means "more than"
"at least" means greater than or equal to, >. < means "less than"
"at least" means greater than or equal to, >. Nice Work.
"at least" means greater than or equal to, >. < means "less than or equal to" or "no more than."
For -3|-6| > 18, which statement is true?
- -3|-6| > 18 is false because -3|-6| equals 18 since negative times negative is positive.
- -3|-6| > 18 is false because -3|-6| > 18 since negative number times an absolute value is negative.
- -3|-6| > 18 is false because -3|-6| < 18 since negative number times an absolute value is negative and -18 is less than 18.
- -3|-6| > 18 is true
-3|-6| > 18 is false because |-6| = 6 and (-3)(6) = -18 which is less than 18
-3|-6| > 18 is false because |-6| = 6 and (-3)(6) = -18 which is less than 18
Nice Work. |-6| = 6 and (-3)(6) = -18 which is less than 18
-3|-6| > 18 is false because |-6| = 6 and (-3)(6) = -18 which is less than 18
Which inequality matches this graph?
- y > 3x
- y < 3x
- y ≥ 3x
- y ≤ 3x
Shading below a solid line means "less than or equal to" < not greater than
Shading below a solid line means "less than or equal to" < not just less than
Shading below a solid line means "less than or equal to" < not greater than or equal to
Shading below a solid line means "less than or equal to" <. Nice Work.
Umio wanted to by apples for 20 cents each and bananas for 15 cents each. She could spend less than $1.50. Let A = apples and B = bananas. The inequality is 0.20A + 0.15B < 1.50 The graph below represents the relationship between the apples and oranges bought by Umio.
What does the point (2,4) in the shaded region mean?
- The point (2,4) is a solution.
- The point (2,4) is not a solution.
- The point (2,4) is the only solution.
You are right here. The point (2,4) is a solution because it lies in the shaded region. It means that Umio can buy 2 apples and 2 oranges. The cost for the apples would be (2 x .20) = $0.40. The cost for the oranges would be (4 x .15) = $0.60. The total for the apples and the oranges is $0.40 + $0.60 = $1.00 which is less than $1.50.
The point (2,4) is a solution because it lies in the shaded region. It means that Umio can buy 2 apples and 2 oranges. The cost for the apples would be (2 x .20) = $0.40. The cost for the oranges would be (4 x .15) = $0.60. The total for the apples and the oranges is $0.40 + $0.60 = $1.00 which is less than $1.50.
The point (2,4) is a solution because it lies in the shaded region. It means that Umio can buy 2 apples and 2 oranges. The cost for the apples would be (2 x .20) = $0.40. The cost for the oranges would be (4 x .15) = $0.60. The total for the apples and the oranges is $0.40 + $0.60 = $1.00 which is less than $1.50.
What does the graph below tell us about how the original equation y = |x| was transformed?
y > |x + 2| + 3
- The original graph was shifted left 2 spaces and up 3 spaces.
- The original graph was shifted right 2 spaces and down 3 spaces.
- The original graph was shifted left 3 spaces and up 2 spaces.
- The original graph was shifted right 3 spaces and down 2 spaces.
You are correct! The original graph was shifted left 2 spaces and up 3 spaces.
Since |x+2| is an absolute value we should see a "v" shape. The +2 inside the absolute value tells us the graph shifts left 2 spaces. The +3 as the constant shifts the graph up 3 spaces.
Since |x+2| is an absolute value we should see a "v" shape. The +2 inside the absolute value tells us the graph shifts left 2 spaces. The +3 as the constant shifts the graph up 3 spaces.
Since |x+2| is an absolute value we should see a "v" shape. The +2 inside the absolute value tells us the graph shifts left 2 spaces. The +3 as the constant shifts the graph up 3 spaces.
Summary
Questions answered correctly:
Questions answered incorrectly:
Inequality Signs
| > means greater than | Elephant's size > Mouse's size | An elephant is bigger than a mouse. |
| < means less than | length of your foot < length of your arm | Your foot isn't as long as your arm. |
| ≥ means greater than or equal to (or "at least") | 5 ≥ 5 | The number 5 is greater than or equal to 5. (in this case, it's equal to 5) |
| ≤ means less than or equal to (or "no more than") | Part time hours ≤ 37.5 hours | Your boss can let your work no more than 37.5 hours a week as a part-time worker. Any more hours will make you full-time. |
Inequalities Practice
Let's practice writing inequalities. Choose the correct symbol for the items below.
| 15 ? 20 | 15 < 20. The point goes toward the smaller amount. |
| 7x is no more than 21 | 7x ≤ 21. 7x is equal to or less than 21. |
| 3x is less than or equal to -6 | 3x ≤ -6 |
| 9y is greater than 53x + 21 | 9y > 53x + 21 |
Graphing Inequalities
To graph inequalities, follow these tips.
- If there is only one variable, use a number line.
- If the inequality uses > or <, use an open circle and arrow on the number line.
- If the inequality uses > or <, use a closed circle to represent "equal to".
x > 4
x ≥ 4
Inequalities with Two Variables
To graph inequalities with two variables, like y < 3x + 5, on the coordinate plane, do the following:
- If the inequality uses > or <, use a dotted line on the graph.
- If the inequality uses ≥ or ≤, use a solid line on the graph to represent "equal to".
- Shade above the line on the graph for > or ≥.
- Shade below the line for < or ≤.
Example
y < 3x + 6
Calculate the x and y-intercepts
| x = 0 | y < 3(0) + 6 y < 6 |
(0, 6) |
| y = 0 | 0 < 3x + 6 3x < -6 x < -2 |
(-2, 0) |
Draw the best-fit dotted line on the graph and shade below the dotted line since we have <.
Inequality Graph Example
Graph 3x + 6y ≥ 18
Calculate the x and y-intercepts: (0, 3) and (6, 0). The best-fit line will be solid and shaded above because ≥ is "greater than or equal to."
Inequality Graph Practice
Describe the graph for the following inequality. List the x and y intercepts, whether the line is dotted or solid, and whether the shading goes above or below the line. Click the Answer button to check your answers and see the graph.
2x + 4y ≤ 20
The x and y intercepts are (0, 5) and (10, 0).
The line is solid.
The shading is below the line.
Absolute Value
Absolute Value is the quantity that represents the distance from zero on the number line. The absolute value is always positive.
Examples
| |5| = 5 | |-5| = 5 |
| |x − 8| = 3 when x = 5 | |5 − 8| = |-3| = 3 |
Be careful, though. If the negative sign is outside the absolute value symbol, make sure you keep it.
| -3|-5| = -3(5) = -15 | -3|5| = -3(5) = -15 |
Graphing Absolute Value
The graph of an absolute value looks like a V or an upside down V.
y = |x|
| x | |x| | y |
| -5 | 5 | 5 |
| -3 | 3 | 3 |
| 0 | 0 | 0 |
| 3 | 3 | 3 |
| 5 | 5 | 5 |
Graphing Inequalities with Absolute Values
The same rules apply for graphing inequalities involving absolute values. Dotted lines for > or <. Solid lines for ≤ or ≥. Shade above for greater than or equal (≥) or greater than (>). Shade below for less than or equal to (≤) or for less than (<).
Practice Graphing Inequalities with Absolute Value
Describe the graph for this inequality. List the x and y intercepts, whether the line is dotted or solid, whether the shading is above or below the line, and whether the graph is a V or upside down V. Click the Answer button when you're finished to check your work and see the graph.
y > |x| + 3
The x and y intercepts are (0, 3) and (-3, 0).
The line is solid.
The shading is below the line.
The shape is a V.
| x | |x| | y |
| -5 | 5 | 8 |
| -3 | 3 | 6 |
| 0 | 0 | 3 |
| 3 | 3 | 6 |
| 5 | 5 | 8 |
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Now that you have had some more practice, you should be better prepared for your quiz. If you still do not feel confident about any topics, please contact your teacher for some additional help.
