Find the value of p in the inequality.
three fourths times p plus 8 is greater than or equal to 3
a) p is greater than or equal to negative twenty over three
b) p is less than or equal to negative twenty over three
c) p is greater than or equal to negative forty four over three
d) p is less than or equal to negative forty four over three
To solve the inequality, we need to isolate p.
Starting with the given inequality: 3/4p + 8 ≥ 3
Subtracting 8 from both sides: 3/4p ≥ -5
To get p alone, we need to divide both sides by 3/4:
p ≥ -5 / (3/4)
To divide by a fraction, we multiply by its reciprocal. The reciprocal of 3/4 is 4/3.
p ≥ -5 * (4/3)
Simplifying:
p ≥ -20/3
Thus, the value of p in the inequality is greater than or equal to -20/3.
Therefore, the correct option is:
a) p is greater than or equal to negative twenty over three
Determine the value of y in the inequality.
18 + 3y < 27
a) y > 3
b) y < 3
c) y > −9
d) y < −9
To solve the inequality, we need to isolate y.
Starting with the given inequality: 18 + 3y < 27
Subtracting 18 from both sides: 3y < 9
Dividing both sides by 3: y < 3
Thus, the value of y in the inequality is less than 3.
Therefore, the correct option is:
b) y < 3
Solve for n.
−2(n − 3.5) > 15
a) n > −4
b) n < −4
c) n > 4
d) n < 4
To solve the inequality, we need to isolate n.
Starting with the given inequality: −2(n − 3.5) > 15
Distributing the -2 to both terms inside the parentheses: -2n + 7 > 15
Subtracting 7 from both sides: -2n > 8
Dividing both sides by -2 and flipping the inequality sign since we are dividing by a negative number: n < -4
Thus, the value of n that satisfies the inequality is n < -4.
Therefore, the correct option is:
b) n < -4
Which graph represents the solution to the inequality 2(b + 2) ≥ 24?
a) number line with open point at 10 with arrow pointing left
b) number line with closed point at 10 with arrow pointing left
c) number line with closed point at 10 with arrow pointing right
d) number line with open point at 10 with arrow pointing right
To solve the inequality, we need to isolate b.
Starting with the given inequality: 2(b + 2) ≥ 24
Expanding the expression: 2b + 4 ≥ 24
Subtracting 4 from both sides: 2b ≥ 20
Dividing both sides by 2: b ≥ 10
Thus, the solution to the inequality is b greater than or equal to 10.
On a number line, if we place a closed point at 10, with an arrow pointing to the right, it represents all values of b that are greater than or equal to 10.
Therefore, the correct option is:
c) number line with closed point at 10 with arrow pointing right
The inequality two thirds minus one half times p is less than or equal to eight thirds is given.
Part A: Solve the inequality for p. Show each step of your work.
Part B: How would you graph your solution to Part A on a number line? Explain in words.
Part A:
Given inequality:
2/3 - 1/2 * p ≤ 8/3
To solve this inequality for p, let's start by subtracting 2/3 from both sides:
-1/2 * p ≤ 8/3 - 2/3
Simplifying the right side:
-1/2 * p ≤ 6/3
Next, we can simplify the right side further:
-1/2 * p ≤ 2
To isolate p, we now need to divide both sides by -1/2. Remember, when we divide or multiply an inequality by a negative number, we also need to flip the inequality sign:
p ≥ (2) / (-1/2)
Simplifying the right side:
p ≥ 2 * (-2/1)
p ≥ -4
Therefore, the solution to the inequality is p greater than or equal to -4.
Part B:
To graph the solution on a number line, we would mark a closed point at -4 (because the inequality is inclusive of equal values) and draw an arrow pointing to the right. This indicates that all values greater than or equal to -4 satisfy the inequality.
The graph on the number line will look like this:
------●---->
-4
Note: The arrow pointing to the right indicates that the values continue indefinitely to the right on the number line.
To find the value of p in the inequality, "three fourths times p plus 8 is greater than or equal to 3", we will solve the inequality step by step.
1. Start with the given inequality: (3/4) * p + 8 ≥ 3
2. Subtract 8 from both sides of the inequality to isolate the term with p:
(3/4) * p + 8 - 8 ≥ 3 - 8
(3/4) * p ≥ -5
3. Multiply both sides of the inequality by the reciprocal of the fractional coefficient (3/4) to solve for p:
(4/3) * [(3/4) * p] ≥ (4/3) * (-5)
p ≥ -20/3
Therefore, the value of p that satisfies the inequality is greater than or equal to -20/3.
Hence, the correct answer is option a) p is greater than or equal to negative twenty over three.