8th grade Math Lesson 2-8 Practice
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. ax - 4 < 12
To solve for x, we need to isolate the variable on one side of the inequality. Let's start by adding 4 to both sides:
ax - 4 + 4 < 12 + 4
This simplifies to:
ax < 16
Next, divide both sides of the inequality by a:
(ax)/a < 16/a
x < 16/a
The solution is x < 16/a.
To graph this solution on a number line, we will use an open circle to represent the solution and shade the line to the left of the circle to indicate that x is less than the value.
2. -4(4 + 7x) ≥ -7x + 5
Start by simplifying the left side:
-4(4 + 7x) = -16 - 28x
The inequality becomes:
-16 - 28x ≥ -7x + 5
To solve for x, let's first bring all terms with x to one side of the inequality:
-16 - 28x + 7x ≥ 5
Combine like terms:
-16 - 21x ≥ 5
Now, let's isolate x by adding 16 to both sides:
-16 + 16 - 21x ≥ 5 + 16
Simplifying further:
-21x ≥ 21
Divide both sides by -21 (remember to flip the inequality when dividing by a negative number):
x ≤ -1
The solution is x ≤ -1.
To graph this solution on a number line, we will use a closed circle to represent the solution and shade the line to the left of the circle to indicate that x is less than or equal to the value.
3. 2(2x - 8) - 8x ≤ 0
Start by simplifying the left side:
2(2x - 8) - 8x = 4x - 16 - 8x = -4x - 16
The inequality becomes:
-4x - 16 ≤ 0
To solve for x, let's first add 16 to both sides:
-4x - 16 + 16 ≤ 0 + 16
Simplifying further:
-4x ≤ 16
Next, divide both sides by -4 (remember to flip the inequality when dividing by a negative number):
x ≥ -4
The solution is x ≥ -4.
To graph this solution on a number line, we will use a closed circle to represent the solution and shade the line to the right of the circle to indicate that x is greater than or equal to the value.
4. 5x + (x - 8) > (9/2)(2x - 4)
Start by simplifying the left side:
5x + x - 8 > 9x - 18
Combine like terms:
6x - 8 > 9x - 18
To solve for x, let's first subtract 6x and add 18 to both sides:
6x - 8 - 6x + 18 > 9x - 18 - 6x + 18
Simplifying further:
10 > 3x
Now, divide both sides by 3:
10/3 > x
The solution is x < 10/3.
To graph this solution on a number line, we will use an open circle to represent the solution and shade the line to the left of the circle to indicate that x is less than the value.
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?
Let's set up an inequality to represent this situation:
125 + 3x ≤ 200
To solve for x, let's first subtract 125 from both sides:
125 - 125 + 3x ≤ 200 - 125
Simplifying further:
3x ≤ 75
Next, divide both sides by 3:
x ≤ 75/3
Simplifying further:
x ≤ 25
Tom can play basketball no more than 25 times if he wants to spend no more than $200.
1. To solve the inequality ax - 4 < 12, we can start by isolating the variable on one side of the inequality.
Adding 4 to both sides, we get:
ax - 4 + 4 < 12 + 4
This simplifies to:
ax < 16
To solve for x, we need to divide both sides of the inequality by a:
(ax)/a < 16/a
This simplifies to:
x < 16/a
Therefore, the solution is x < 16/a.
To graph this on a number line, plot an open circle at 16/a and shade everything to the left of that point.
2. To solve the inequality -4(4 + 7x) ≥ -7x + 5, we can start by expanding and simplifying both sides of the inequality.
-4 * 4 - 4 * 7x ≥ -7x + 5
Simplifying further:
-16 - 28x ≥ -7x + 5
Next, we can simplify the expression by combining like terms.
-28x + 7x ≥ 5 + 16
Simplifying further:
-21x ≥ 21
To solve for x, we divide both sides of the inequality by -21. Remember that when dividing or multiplying by a negative number, we need to reverse the inequality sign.
x ≤ 21 / -21
This simplifies to:
x ≤ -1
Therefore, the solution is x ≤ -1.
To graph this on a number line, plot a closed circle at -1 and shade everything to the left of that point.
3. To solve the inequality 2(2x - 8) - 8x ≤ 0, we can start by simplifying both sides of the inequality.
2 * 2x - 2 * 8 - 8x ≤ 0
Simplifying further:
4x - 16 - 8x ≤ 0
Combine like terms:
-4x - 16 ≤ 0
To solve for x, we need to isolate the variable.
-4x ≤ 16
Divide both sides of the inequality by -4. Remember to reverse the inequality sign when dividing or multiplying by a negative number.
x ≥ 16/-4
This simplifies to:
x ≥ -4
Therefore, the solution is x ≥ -4.
To graph this on a number line, plot a closed circle at -4 and shade everything to the right of that point.
4. To solve the inequality 5x + (x - 8) > (9/2) * (2x - 4), we can start by simplifying both sides of the inequality.
5x + x - 8 > (9/2)(2x - 4)
Combine like terms:
6x - 8 > (9/2)(2x - 4)
Next, distribute the (9/2) on the right side of the inequality:
6x - 8 > (9/2)(2x) - (9/2)(4)
Simplifying further:
6x - 8 > 9x - 18
To solve for x, we need to isolate the variable.
Subtract 6x from both sides:
-8 > 9x - 6x - 18
Combine like terms:
-8 > 3x - 18
Add 18 to both sides:
10 > 3x
To solve for x, divide both sides of the inequality by 3:
10/3 > x
This simplifies to:
x < 10/3
Therefore, the solution is x < 10/3.
To graph this on a number line, plot an open circle at 10/3 and shade everything to the left of that point.
5. To determine how many times Tom can play basketball if he wants to spend no more than $200, we can set up an equation to solve for the maximum number of games.
Let's assume the number of games Tom can play be represented by g.
The total cost for the membership and renting the basketball courts would be:
$125 (membership fee) + $3g (rental fee per game)
According to the problem, Tom wants to spend no more than $200.
Therefore, we can set up the equation:
$125 + $3g ≤ $200
Subtract $125 from both sides:
$3g ≤ $200 - $125
Simplifying further:
$3g ≤ $75
To solve for g, divide both sides of the inequality by $3:
($3g)/$3 ≤ $75/$3
Simplifying further:
g ≤ 25
Therefore, Tom can play basketball a maximum of 25 times if he wants to spend no more than $200.