8th grade Math Lesson 2-8 Practice
Vocabulary Terms: Define/Illustrate
1. Inequality symbols
2. Closed or open circle in solution set
3. Inequality solving exception
Problems: Show All Work: Solve and graph the solution on a number line.
1.ax - 4 < 12
2.-4(4 + 7x) ≥ -7x + 5
3.2(2x - 8) - 8x ≤ 0
4.5x + (x - 8) > 9/2 (2x - 4)
5.Tom is deciding whether or not he should become a member of the gym to use their basketball courts. The membership cost is $125. Members pay $3 to rent out the basketball courts. How many times can Tom play basketball if he wants to spend no more than $200?
1. Inequality symbols: Inequality symbols are symbols used to compare two quantities. The symbols include > (greater than), < (less than), ≥ (greater than or equal to), ≤ (less than or equal to), and ≠ (not equal to).
Illustration:
Example: 4 > 2 means that 4 is greater than 2.
2. Closed or open circle in solution set: In graphing the solution set of an inequality on a number line, a closed circle is used to represent values that are included in the solution, while an open circle is used to represent values that are not included in the solution.
Illustration:
Example: For the inequality x > 3, an open circle is placed at 3, indicating that 3 is not included in the solution set.
3. Inequality solving exception: An inequality solving exception occurs when the direction of the inequality is reversed when multiplying or dividing both sides of the inequality by a negative number.
Illustration:
Example: When solving the inequality -2x > 8, dividing both sides by -2 would result in a reversal of the inequality symbol, giving x < -4 as the solution.
1. Inequality symbols: Inequality symbols are used to represent relationships between two quantities that are not equal. The most common inequality symbols are less than (<), greater than (>), less than or equal to (≤), and greater than or equal to (≥). For example, if we have an inequality symbol like x < 5, it means that x is any value less than 5.
2. Closed or open circle in solution set: When graphing the solution set of an inequality on a number line, we use closed or open circles to represent whether the endpoint is included or not included in the solution. A closed circle (●) is used when the endpoint is included in the solution, and an open circle (○) is used when the endpoint is not included in the solution. For example, if we have an inequality like x ≤ 3, we would use a closed circle at 3 to indicate that 3 is included in the solution.
3. Inequality solving exception: Inequality solving exceptions occur when there are multiplication or division by a negative number. In these cases, the direction of the inequality symbol must be reversed. For example, if we have an inequality like -2x < 8, we need to divide both sides by -2. However, when dividing by a negative number, the inequality symbol must be reversed, so the solution would be x > -4.
Now let's solve the problems:
1. ax - 4 < 12
To solve this inequality, we'll first isolate the variable, ax.
ax < 16
Next, we'll divide both sides of the inequality by 'a', making sure to consider the sign of 'a'. If 'a' is positive, the inequality symbol remains the same. If 'a' is negative, the inequality symbol is reversed.
If 'a' > 0, then the solution is x < 16/a.
If 'a' < 0, then the solution is x > 16/a.
Remember to graph the solution on a number line using an open or closed circle based on the inequality symbol.
2. -4(4 + 7x) ≥ -7x + 5
First, simplify the equation:
-16 - 28x ≥ -7x + 5
Next, simplify further by combining like terms:
-16 - 28x + 7x ≥ 5
Combine like terms:
-16 - 21x ≥ 5
To isolate the variable, subtract -16 from both sides:
-21x ≥ 21
Finally, divide both sides by -21 and remember to reverse the inequality symbol because we divided by a negative number:
x ≤ -1
Graph the solution on a number line using a closed circle at -1.
3. 2(2x - 8) - 8x ≤ 0
First, distribute the 2:
4x - 16 - 8x ≤ 0
Simplify by combining like terms:
-4x - 16 ≤ 0
To isolate the variable, add 16 to both sides:
-4x ≤ 16
Finally, divide both sides by -4, remembering to reverse the inequality symbol because we divided by a negative number:
x ≥ -4
Graph the solution on a number line using a closed circle at -4.
4. 5x + (x - 8) > (9/2)(2x - 4)
First, distribute the (9/2):
5x + x - 8 > 9x - 18
Combine like terms:
6x - 8 > 9x - 18
To isolate the variable, subtract 9x from both sides:
-3x - 8 > -18
Next, add 8 to both sides:
-3x > -10
Finally, divide both sides by -3, remembering to reverse the inequality symbol because we divided by a negative number:
x < 10/3
Graph the solution on a number line using an open circle at 10/3.
5. To determine how many times Tom can play basketball, we need to set up an inequality. Let's say he plays basketball 'x' times.
The membership cost is $125, and he needs to pay $3 each time he plays basketball. Since he wants to spend no more than $200, we can write the inequality:
125 + 3x ≤ 200
To solve this inequality, begin by subtracting 125 from both sides:
3x ≤ 75
Next, divide both sides by 3:
x ≤ 25
Therefore, Tom can play basketball 25 times or less and still spend no more than $200.