1.) Which of the following inequalities is true for all real values of x?
a.) x^3≥^2
b.) 4x^2≥2x^2
c.) (3x)^2≥4x^2
d.) 2(x-3)^2≥2x^2-3
2.) Which following inequalities is true for all real values of x?
a.) x^5≥x^4
b.) 5x^2≥2x^3
c.) (5x)^2≥4x^2
d.) 3(x-2)^2≥3x^2
clearly, it's b and c
try dividing out the x^2 factor
1.) To determine which of the inequalities is true for all real values of x, let's go through each option:
a.) x^3 ≥ x^2
This inequality is not true for all real values of x. For example, if x = -1, then (-1)^3 = -1 which is less than (-1)^2 = 1. Therefore, option a) is not true for all real values of x.
b.) 4x^2 ≥ 2x^2
This inequality is true for all real values of x. If we simplify both sides, we have 4x^2 ≥ 2x^2, which simplifies to 2x^2 ≥ 0. Since the square of any real number is non-negative, this inequality is true for all real values of x. Therefore, option b) is the correct answer.
c.) (3x)^2 ≥ 4x^2
This inequality is true for all real values of x. If we simplify both sides, we have 9x^2 ≥ 4x^2. Since 9x^2 is always greater than or equal to 4x^2 for all real values of x, this inequality is true. Therefore, option c) is also a correct answer.
d.) 2(x-3)^2 ≥ 2x^2 - 3
This inequality is not true for all real values of x. If we expand and simplify both sides, we have 2x^2 -12x + 18 ≥ 2x^2 - 3. Simplifying further, we get -12x + 18 ≥ -3. However, if we divide by -12, we need to reverse the inequality, giving us x ≤ -15/4. Therefore, option d) is not true for all real values of x.
2.) Let's analyze each option:
a.) x^5 ≥ x^4
This inequality is not true for all real values of x. For example, if x = -1, then (-1)^5 = -1 which is less than (-1)^4 = 1. Therefore, option a) is not true for all real values of x.
b.) 5x^2 ≥ 2x^3
This inequality is not true for all real values of x. For example, if x = 0, then 5(0)^2 = 0 which is less than 2(0)^3 = 0. Therefore, option b) is not true for all real values of x.
c.) (5x)^2 ≥ 4x^2
This inequality is true for all real values of x. If we simplify both sides, we have 25x^2 ≥ 4x^2. Since 25x^2 is always greater than or equal to 4x^2 for all real values of x, this inequality is true. Therefore, option c) is the correct answer.
d.) 3(x-2)^2 ≥ 3x^2
This inequality is not true for all real values of x. If we expand and simplify both sides, we have 3x^2 - 12x + 12 ≥ 3x^2. Simplifying further, we get -12x + 12 ≥ 0. However, if we divide by -12, we need to reverse the inequality, giving us x ≤ -1. Therefore, option d) is not true for all real values of x.
Therefore, the correct answer for question 2 is option c), (5x)^2 ≥ 4x^2.
To determine which inequalities are true for all real values of x, we can solve each inequality and analyze the results.
1.) For the inequality x^3 ≥ x^2:
To solve this inequality, we need to find the values of x that satisfy the inequality.
Step 1: Subtract x^2 from both sides:
x^3 - x^2 ≥ 0
Step 2: Factor out x^2:
x^2(x - 1) ≥ 0
Step 3: Analyze the sign of each factor.
For x^2, it will always be non-negative or zero for all real values of x.
For (x - 1), it will change sign at x = 1.
Step 4: Determine the sign of the product.
When both factors have the same sign (either both positive or both negative), the product will be positive or zero. Therefore, this inequality is true for all values of x.
2.) For the inequality 4x^2 ≥ 2x^2:
To solve this inequality, we need to find the values of x that satisfy the inequality.
Step 1: Subtract 2x^2 from both sides:
4x^2 - 2x^2 ≥ 0
Step 2: Simplify:
2x^2 ≥ 0
Step 3: Analyze the sign of the expression.
2x^2 will always be non-negative or zero for all real values of x.
Step 4: Determine the sign of the expression.
Since the expression is always non-negative, this inequality is true for all values of x.
3.) For the inequality (3x)^2 ≥ 4x^2:
To solve this inequality, we need to find values of x that satisfy the inequality.
Step 1: Expand and simplify:
9x^2 ≥ 4x^2
Step 2: Subtract 4x^2 from both sides:
5x^2 ≥ 0
Step 3: Analyze the sign of the expression.
5x^2 will always be non-negative or zero for all real values of x.
Step 4: Determine the sign of the expression.
Since the expression is always non-negative, this inequality is true for all values of x.
4.) For the inequality 2(x-3)^2 ≥ 2x^2 - 3:
To solve this inequality, we need to find values of x that satisfy the inequality.
Step 1: Expand and simplify:
2(x^2 - 6x + 9) ≥ 2x^2 - 3
Step 2: Distribute:
2x^2 - 12x + 18 ≥ 2x^2 - 3
Step 3: Subtract 2x^2 from both sides:
-12x + 18 ≥ -3
Step 4: Subtract 18 from both sides:
-12x ≥ -21
Step 5: Divide by -12 (Note: when dividing by a negative number, the direction of the inequality changes):
x ≤ 21/12 or x ≤ 7/4
However, the inequality states "greater than or equal to." Therefore, this inequality is not true for all real values of x.
Now let's move to the second set of inequalities:
1.) For the inequality x^5 ≥ x^4:
To solve this inequality, we need to find values of x that satisfy the inequality.
Step 1: Subtract x^4 from both sides:
x^5 - x^4 ≥ 0
Step 2: Factor out x^4:
x^4 (x - 1) ≥ 0
Step 3: Analyze the sign of each factor.
For x^4, it will always be non-negative or zero for all real values of x.
For (x - 1), it will change sign at x = 1.
Step 4: Determine the sign of the product.
When both factors have the same sign (either both positive or both negative), the product will be positive or zero. Therefore, this inequality is true for all values of x.
2.) For the inequality 5x^2 ≥ 2x^3:
To solve this inequality, we need to find values of x that satisfy the inequality.
Step 1: Subtract 5x^2 from both sides:
0 ≥ 2x^3 - 5x^2
Step 2: Analyze the expression.
There is no obvious way to solve this inequality algebraically, so we can use a different approach.
Step 3: Graph the function f(x) = 2x^3 - 5x^2:
By graphing the function or using a graphing calculator, we can observe the behavior of the function and determine the range of x values for which the inequality is satisfied.
After graphing the function, we find that the graph is below the x-axis when x < 0 and above the x-axis when x > 0. Therefore, the inequality 5x^2 ≥ 2x^3 is not true for all real values of x.
3.) For the inequality (5x)^2 ≥ 4x^2:
To solve this inequality, we need to find values of x that satisfy the inequality.
Step 1: Expand and simplify:
25x^2 ≥ 4x^2
Step 2: Subtract 4x^2 from both sides:
21x^2 ≥ 0
Step 3: Analyze the sign of the expression.
21x^2 will always be non-negative or zero for all real values of x.
Step 4: Determine the sign of the expression.
Since the expression is always non-negative, this inequality is true for all values of x.
4.) For the inequality 3(x-2)^2 ≥ 3x^2:
To solve this inequality, we need to find values of x that satisfy the inequality.
Step 1: Expand and simplify:
3(x^2 - 4x + 4) ≥ 3x^2
Step 2: Distribute:
3x^2 - 12x + 12 ≥ 3x^2
Step 3: Subtract 3x^2 from both sides:
-12x + 12 ≥ 0
Step 4: Divide by 12:
- x + 1 ≥ 0
Step 5: Multiply both sides by -1 (Note: when multiplying or dividing by a negative number, the direction of the inequality changes):
x - 1 ≤ 0 or x ≤ 1
However, the inequality states "greater than or equal to." Therefore, this inequality is not true for all real values of x.
In summary:
- For the first set of inequalities, the inequality c.) (3x)^2 ≥ 4x^2 is true for all real values of x.
- For the second set of inequalities, the inequality a.) x^5 ≥ x^4 is true for all real values of x.