Example Question - inequalities
Here are examples of questions we've helped users solve.
Completing Inequalities with Cube Root of 107
The image shows an incomplete mathematical statement involving an inequality with the cube root of 107, with two empty boxes indicating missing values that should be placed to make the inequality correct. To find the numbers that fit into these boxes, we need to consider the cube root of 107. Since \(4^3 = 64\) and \(5^3 = 125\), the cube root of 107 must be between 4 and 5 because 107 is between 64 and 125. Now the statement should read as follows to make the inequalities true: \[4 < \sqrt[3]{107} < 5\] So the numbers that fit into the boxes are 4 and 5.
Finding Numbers Related to Square Root of 2
The image displays two empty boxes separated by inequality signs with the square root of 2 in the middle, like so: \[ \Box < \sqrt{2} < \Box \] We are likely being asked to find two numbers that satisfy these inequalities, where one number is less than the square root of 2 and the other is greater. The square root of 2 is an irrational number which is approximately 1.414. So we need to find a number that is less than 1.414 and another that is greater. A simple solution is to use 1 for the first box and 2 for the second box, as 1 is less than the square root of 2 and 2 is greater than square root of 2. The resulting completed inequality would look like this: \[ 1 < \sqrt{2} < 2 \] This is a valid solution since 1 and 2 are integers that satisfy the inequalities on either side of the square root of 2.
Understanding Inequalities Around -43
The image shows an inequality with two blank rectangles and a number in between, written as "< -43 <". To solve it, you need to insert mathematical symbols to make the statement true. Since -43 is a negative number, to make the inequality true, you'd want to indicate that the numbers decreasing from -43 are on the left and the numbers increasing from -43 are on the right. This means that you would use the "greater than" symbol, ">", on the left and the "less than" symbol, "<", on the right to show that the numbers on the left are less than -43 and the numbers on the right are greater than -43. To correctly fill in the blanks, the inequality should read: "< -43 >" This expresses that -43 is greater than the numbers to its left and less than the numbers to its right. However, this statement doesn't follow conventional notation because usually the "less than" symbol point towards the smaller number, making such a completed statement generally incorrect. A correct mathematical statement that reflects the ordering of numbers around -43 would be: "... < -43 < ..." This means any number to the left of -43 is less than it, and any number to the right is greater, which corresponds to the standard number line orientation.
Understanding Complex Numbers and Inequalities
The image shows an inequality with a square root of a negative number: the square root of -86. In the real number system, the square root of a negative number is not defined because no real number squared gives a negative result. However, in the complex number system, the square root of a negative number involves the imaginary unit \( i \), where \( i^2 = -1 \). To express the square root of -86, we can factor out the imaginary unit \( i \), resulting in: \[ \sqrt{-86} = \sqrt{-1 \cdot 86} = \sqrt{-1} \cdot \sqrt{86} = i \sqrt{86} \] Since you're asked to place the square root of -86 within inequalities, it's important to note that complex numbers do not have a natural ordering like real numbers, so you cannot say that one complex number is greater than or less than another. Thus, the image prompts an operation which is not valid within the real number system and cannot be completed as a typical inequality. If we were to attempt to place this value in an inequality with real numbers, we could not do so meaningfully, as the complex number cannot be directly compared to real numbers in terms of being greater or lesser. However, the absolute value or magnitude of a complex number can be compared to real numbers. The magnitude of \( i\sqrt{86} \) is \( \sqrt{86} \), and we know that real numbers less than \( \sqrt{86} \) would be to the left of it on the real number line, and numbers greater than \( \sqrt{86} \) would be to the right if we were considering magnitude alone, ignoring the imaginary component. But without more context, and strictly speaking, the comparison symbols (<) in the image are not meaningful when applied to a complex number.
Understanding Complex Numbers and Inequalities
The expression in the image includes the square root of a negative number: \(-\sqrt{86}\). In the realm of real numbers, you cannot take the square root of a negative number, as square roots are defined only for non-negative numbers. However, in the context of complex numbers, the square root of a negative number is a multiple of the imaginary unit \(i\), where \(i\) is defined as \(i^2 = -1\). Thus, \(-\sqrt{86}\) in terms of complex numbers is \(-i\sqrt{86}\). This number is purely imaginary, and there is no real number less than or greater than it because comparisons of greater or less than do not apply to imaginary numbers. Therefore, the blank boxes on both sides of \(-\sqrt{86}\) cannot be filled with real numbers to create a true statement about inequalities. Filling these blanks would require a context where complex numbers are included, and even then, the notion of "greater than" or "less than" isn't meaningful in the same way as it is with real numbers.
Solving Inequalities with Minimum Integer
题目给出的不等式是 `-3 < 9 - 2p < 4`。我们需要找到满足这个不等式的最小整数`p`。 首先，我们将不等式分为两部分处理： 1. `-3 < 9 - 2p` 2. `9 - 2p < 4` 对第一个不等式 `-3 < 9 - 2p` 进行操作，得到： `-3 - 9 < -2p` `-12 < -2p` `6 > p` (两边同时除以 `-2` 并且翻转不等号，因为我们是除以负数) 对第二个不等式 `9 - 2p < 4` 进行操作，得到： `-2p < 4 - 9` `-2p < -5` `p > 2.5` (两边同时除以 `-2` 并且翻转不等号) 结合两个部分中间的 `p`，我们得到 `2.5 < p < 6`。所以满足条件的最小整数 `p` 是 `3`。 答案是 B) `3`。
Solving Equations in the Provided Image
Tôi sẽ hướng dẫn bạn giải các phương trình trong hình ảnh bạn cung cấp. Chúng ta sẽ làm từng câu một. **Câu 1:** Giải các bất phương trình sau: a) \(2x^2 - 5x + 2 < 0\) Ta tìm nghiệm của phương trình \(2x^2 - 5x + 2 = 0\). Áp dụng công thức nghiệm của phương trình bậc hai: \(a = 2\), \(b = -5\), \(c = 2\) Delta (\(\Delta\)) được tính bằng \(b^2 - 4ac = (-5)^2 - 4 \cdot 2 \cdot 2 = 25 - 16 = 9\) Nghiệm của phương trình là: \(x_1 = \frac{-b + \sqrt{\Delta}}{2a} = \frac{5 + 3}{4} = 2\) \(x_2 = \frac{-b - \sqrt{\Delta}}{2a} = \frac{5 - 3}{4} = \frac{1}{2}\) Vì hệ số \(a = 2 > 0\) nên parabol hướng lên trên. Như vậy, bất phương trình \(2x^2 - 5x + 2 < 0\) có nghiệm là \(x \in \left( \frac{1}{2}, 2 \right)\). b) \(\sqrt{x + 2} - 12 = 8 - x\) Đặt điều kiện xác định (ĐKXĐ) cho căn thức bên trái là \(x + 2 \geq 0 \Leftrightarrow x \geq -2\) Giải phương trình: \(\sqrt{x + 2} = 20 - x\) Bình phương hai vế của phương trình ta được: \(x + 2 = (20 - x)^2\) Tiếp tục giải phương trình bậc hai này: \(x^2 - 40x + 398 = 0\) Sử dụng công thức nghiệm ta có: Delta (\(\Delta\)) = \(b^2 - 4ac = (-40)^2 - 4 \cdot 1 \cdot 398 = 1600 - 1592 = 8\) Do \(\Delta > 0\) nên phương trình có hai nghiệm phân biệt: \(x_1 = \frac{40 + \sqrt{8}}{2} = 20 + \sqrt{2}\) \(x_2 = \frac{40 - \sqrt{8}}{2} = 20 - \sqrt{2}\) Kiểm tra lại với ĐKXĐ, cả hai nghiệm đều thỏa mãn. Như vậy, nghiệm của phương trình ban đầu là \(x_1 = 20 + \sqrt{2}\) và \(x_2 = 20 - \sqrt{2}\). **Chú ý:** Phần còn lại của các câu hỏi không được giải trong lần trả lời này. Nếu bạn cần giải các câu hỏi khác, xin vui lòng yêu cầu mỗi lần một câu để có thể giải thích một cách chi tiết và rõ ràng.
Finding Equation of a Line and Inequalities
To find the equation of the line, you need to find its slope and y-intercept. From the graph, you can see the line crosses the y-axis at the point (0, 1), which means the y-intercept is 1. To find the slope, you can pick two points on the line. The line passes through the points (0, 1) and (-2, -1) which can be clearly seen on the graph. The slope (m) is calculated as the rise over the run: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 1}{-2 - 0} = \frac{-2}{-2} = 1. \] So, the slope is 1. With a slope of 1 and a y-intercept of 1, the equation of the line in slope-intercept form (y = mx + b) is: \[ y = 1x + 1 \] or simply: \[ y = x + 1. \] For inequalities: a) Since the line is solid, this means that the inequality includes equality; that is, points on the line satisfy the inequality. Since the area below the line is shaded, the inequality represents all points where y is less than or equal to x + 1. Therefore, the inequality for part a) is: \[ y \leq x + 1. \] b) and c) are not provided explicitly in the question, but typically they would be related to different shading regions if provided. If the shaded region was above the line for part b), for example, the inequality would be y ≥ x + 1. If the shaded region for part c) omitted points on the line, the inequality would use strict inequality signs (< or >) rather than ≤ or ≥.
Equation of a Line and Inequalities in Shaded Regions
I'll help you to determine the equation of the line shown in part a of the image, and then we can use this information to write inequalities for the shaded regions shown in parts b and c. To find the equation of a line, we need to identify two key components: the slope and the y-intercept. In the coordinate grid provided, the line crosses the y-axis at (0, -2), which means the y-intercept is -2. The line also looks as if it passes through another point with integer coordinates, which we can use to determine the slope. I see that the line goes through (2, 2) along with (0, -2). The slope (\(m\)) is the change in y over the change in x when moving from one point to another: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - (-2)}{2 - 0} \] \[ m = \frac{4}{2} = 2 \] So the slope of the line is 2. Now that we have both the slope and the y-intercept, we can write the equation of the line in slope-intercept form (y = mx + b): \[ y = 2x - 2 \] For part b, if the graph is shaded above this line, then the inequality representing the shaded region would use a "greater than" symbol, because the y-values are greater than the y-values on the line for any given x-value. The inequality for the graph in part b would be: \[ y > 2x - 2 \] For part c, if the graph is shaded below this line, then the inequality representing the shaded region would use a "less than" symbol, because the y-values are less than the y-values on the line for any given x-value. The inequality for the graph in part c would be: \[ y < 2x - 2 \]
Determining Equations of Lines and Inequalities
To find the equation of the line in part a, we need to determine the slope and the y-intercept. The equation of a line in slope-intercept form is: y = mx + b where m is the slope of the line, and b is the y-intercept. We can find the slope by looking at two points on the line and using the slope formula: slope (m) = (change in y) / (change in x) From the image, we can pick two points that the line passes through. Let's choose (0, -2) and (2, -1) since they're clearly on the grid intersections. Now, we find the slope: m = (y2 - y1) / (x2 - x1) m = (-1 - (-2)) / (2 - 0) m = (1) / (2) m = 1/2 Next, we find the y-intercept (b). This is where the line crosses the y-axis. Looking at the graph, we can see that this occurs at (0, -2), so b = -2. Now we have the slope m = 1/2 and y-intercept b = -2, the equation of the line is: y = (1/2)x - 2 To find the inequalities for parts b and c, we would normally look for instructions that designate whether the area above or below the line should be shaded. Since the image only shows the line and does not specify the inequalities or shaded regions, we can't determine what the inequalities would be. However, if you are being asked for the inequality that includes points below the line, the inequality would be: y ≤ (1/2)x - 2 If the inequality should include points above the line, it would be: y ≥ (1/2)x - 2 Remember, without additional information or context regarding which side of the line should be considered for the inequality, you cannot definitively state the inequality.
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