Consider the inequality 3( - 1 - x) ≤ -2x - 3x.
The solution set of the inequality is:
(1 point)
Responses
A) {x|x≥32}
{x|xis greater than or equal to 3 halves}
B) {x|x≤32}
{x|xis less than or equal to 3 halves}
C) {x|x≥23}
{x|xis greater than or equal to 2 thirds}
D){x|x≤23}
To solve the inequality, we need to simplify both sides and then solve for x.
Starting with the left side of the inequality, we distribute the 3 to both terms inside the parentheses:
3(-1 - x) = -3 - 3x.
Now let's simplify the right side of the inequality:
-2x - 3x = -5x.
Plugging these simplified expressions back into the original inequality, we have:
-3 - 3x ≤ -5x.
Next, let's get all the x terms on one side of the inequality by adding 5x to both sides:
-3 + 2x ≤ 0.
Now, let's isolate the x term by adding 3 to both sides:
2x ≤ 3.
Finally, divide both sides by 2 to solve for x:
x ≤ 3/2.
So the solution set of the inequality is {x | x ≤ 3/2}.
Therefore, the correct answer is:
B) {x | x ≤ 3/2}.
Graph the previous solution set on a number line.(1 point)
Responses
I'm sorry, but as a text-based AI, I am unable to directly generate graphical representations. However, I can describe how to graph the solution set on a number line.
To graph the solution set {x | x ≤ 3/2} on a number line, you would first draw a horizontal line with a labeled point at 3/2. Then, shade the region to the left of and including the point 3/2, to represent all the values of x that satisfy the inequality x ≤ 3/2.
I apologize for any inconvenience caused.
Solve the inequality: -2(2x - 4) ≤ 4(2 - x).(1 point)
Responses
Let's simplify the inequality step by step.
Starting with the left side of the inequality, we distribute the -2 to both terms inside the parentheses:
-2(2x - 4) = -4x + 8.
Now let's simplify the right side of the inequality:
4(2 - x) = 8 - 4x.
Plugging these simplified expressions back into the original inequality, we have:
-4x + 8 ≤ 8 - 4x.
Next, let's get all the x terms on one side of the inequality by adding 4x to both sides:
8 ≤ 8.
No matter the value of x, the inequality 8 ≤ 8 is always true. Therefore, the solution set for this inequality is all real numbers or (-∞, ∞).
Thus, the correct answer is:
{x | x is a real number} or (-∞, ∞).
Which values are in the solution set of the inequality −23x + 13 ≥ −1 ?
To solve the inequality, we need to isolate the variable x.
Starting with the inequality:
-23x + 13 ≥ -1.
First, let's subtract 13 from both sides:
-23x ≥ -14.
Next, divide both sides by -23. Remember that when dividing by a negative number, the direction of the inequality sign should be flipped:
x ≤ -14/-23.
Simplifying the expression on the right side:
x ≤ 14/23.
So, the values in the solution set of the inequality are all values of x that are less than or equal to 14/23.
Therefore, the correct answer is:
{x | x ≤ 14/23}.
Which values are in the solution set of the inequality −23x + 13 ≥ −1 ?
19
20
21
22
or 23
To determine which values are in the solution set of the inequality −23x + 13 ≥ −1, we need to solve the inequality for x.
Starting with the inequality:
-23x + 13 ≥ -1
We can isolate the variable x by subtracting 13 from both sides:
-23x ≥ -1 - 13
-23x ≥ -14
Next, divide both sides by -23. Remember that when dividing by a negative number, the direction of the inequality sign should be flipped:
x ≤ -14/ -23
Simplifying the expression:
x ≤ 14/ 23
So, the values in the solution set of the inequality are all values of x that are less than or equal to 14/23.
Among the given options, the value that satisfies x ≤ 14/23 is 19.
Therefore, the correct answer is:
19
To solve the inequality 3(-1 - x) ≤ -2x - 3x, we need to simplify and isolate the variable x.
First, let's distribute the 3 to both terms inside the parentheses:
3 * -1 - 3 * x ≤ -2x - 3x
-3 - 3x ≤ -2x - 3x
Next, let's combine like terms on both sides of the inequality:
-3 - 3x ≤ -5x
Now, let's isolate the variable x by bringing all the terms with x to one side of the inequality:
-3 + 5x - 3x ≤ 0
2x - 3 ≤ 0
To solve for x, we need to divide both sides of the inequality by 2:
(2x - 3)/2 ≤ 0 /2
x - 1.5 ≤ 0
Now, we have x - 1.5 ≤ 0. To find the solution set, we need to determine the values of x that satisfy this inequality.
Since the inequality is less than or equal to 0, x can take values that are less than or equal to 1.5.
Therefore, the correct solution set for the inequality 3(-1 - x) ≤ -2x - 3x is:
{x | x ≤ 1.5}
So, the correct answer is:
B) {x | x ≤ 1.5}