Example Question - solving inequality
Here are examples of questions we've helped users solve.
Solving Inequality 5q + 3 < 37
To solve the inequality 5q + 3 < 37, we want to isolate the variable q. We'll do this by first subtracting 3 from both sides of the inequality to eliminate the constant term on the left side: 5q + 3 - 3 < 37 - 3 5q < 34 Next, we'll divide both sides of the inequality by 5 to solve for q: 5q / 5 < 34 / 5 q < 6.8 So we're looking for a value of q that is less than 6.8. Among the options provided: - q = -8 is less than 6.8. - q = -5 is less than 6.8. - q = -2 is less than 6.8. - q = -1 is less than 6.8. All the given options for q are less than 6.8, meaning they all satisfy the inequality 5q + 3 < 37. Therefore, they are all correct solutions to the inequality.
Solving Inequality 5q + 3 < 37
To solve the inequality 5q + 3 < 37, follow these steps: 1. Subtract 3 from both sides to isolate the term with the variable q: 5q + 3 - 3 < 37 - 3 5q < 34 2. Divide both sides by 5 to solve for q: 5q/5 < 34/5 q < 6.8 Now let's check the provided options to determine which one is a solution: - q = -8: This is indeed less than 6.8, so it is a possible solution. - q = -5: This is also less than 6.8, so it is a possible solution. - q = -2: Similarly, this is less than 6.8, so it is a possible solution. - q = -1: This, too, is less than 6.8, so it is a possible solution. All the provided options are less than 6.8. Therefore, all of the given options are solutions to the inequality 5q + 3 < 37.
Solving Absolute Value Inequality in Mathematics
لا يوفر السؤال معلومات كافية لإعطاء إجابة واضحة لأن الصورة لا تظهر السؤال بأكمله. ومع ذلك، يمكنني رؤية جزء من سؤال الرياضيات الذي يحتوي على معادلة: \[ \frac{1}{|x|} > 1 \] لحل هذه المعادلة، يجب علينا معرفة القيم التي تجعل قيمة المقام هذه أصغر من 1. وبما أن المقام هو القيمة المطلقة لـ \( x \)، فإننا نبحث عن القيم التي تجعل \( |x| \) أقل من 1. القيم المطلقة تكون دائماً غير سالبة، لذا نحن نبحث عن جميع \( x \) حيث \( 0 < |x| < 1 \). هذا يعني أن \( -1 < x < 1 \) ولكن لا يمكن أن يكون \( x = 0 \) لأن هذا سيتسبب بجعل المقام غير مُعرَّف (تقسيم على صفر). وبالتالي، الحل هو الفترة المفتوحة (-1, 1) باستثناء الصفر.
Solving an Inequality with Negative Square Root
The image shows an inequality with two empty boxes on either side of the negative square root of 71. To solve this, we need to find two numbers that are less than and greater than the negated square root of 71. Recall that taking the square root of a positive number results in a positive number and when we add a negative sign to it, it makes the expression negative. The square root of 71 is approximately 8.43 (since √64 = 8 and √81 = 9, and 71 is between 64 and 81). Therefore, -√71 is approximately -8.43 Since we need a number less than -8.43 and a number greater than -8.43, we could choose integers for simplicity. For the first box, let's choose -9 because -9 is less than -8.43. For the second box, we could choose -8 because -8 is greater than -8.43. So, in "en" language code, the inequality would be filled in as: -9 < -√71 < -8
Understanding Inequalities with Square Root of 2
The image shows two inequality signs surrounding the square root of 2 (√2), with blanks where numbers would be. To solve this problem, you need to find two numbers that are consecutive integers where the square root of 2 fits between them. The square root of 2 is approximately 1.414. It is more than 1 but less than 2. Therefore, the two integers that would fit on either side of √2 would be 1 and 2. The inequality then reads as: 1 < √2 < 2 This indicates that the square root of 2 is greater than 1 but less than 2.
Solving Inequality with Square Roots
The image shows an inequality with two blank boxes and the square root of 136 in the center. To solve this, we need to find two square numbers that the square root of 136 falls between. Let's identify the nearest perfect squares around 136. Since \( 11^2 = 121 \) and \( 12^2 = 144 \), we can see that 136 falls between these two squares. So, the square root of 136 is more than 11 but less than 12: \( 11 < \sqrt{136} < 12 \) Hence, the numbers that should be in the blank boxes are 11 and 12, read from left to right: \[ 11 < \sqrt{136} < 12 \]
Solving Inequality with Square Root of 5
It looks like the image you've provided contains an inequality with two blank boxes and the square root of 5. It's difficult to determine specifically what is being asked because there's no context or additional instructions. However, I can give you a general idea of what the inequality suggests. The square root of 5 is an irrational number, approximately equal to 2.236. An inequality featuring the square root of 5 between two boxes might be asking for you to identify two integers, one that is less than the square root of 5, and one that is greater. Given this, the answers could be: - The integer less than √5: 2 - The integer greater than √5: 3 Therefore, the completed inequality with the integers in place would be: 2 < √5 < 3 These integers are the largest whole number less than √5 and the smallest whole number greater than √5, respectively.
Solving Inequality with Square Root
The given image depicts an inequality with a square root in the center: two blank boxes with less than signs around the square root of 104, written as "< √104 <". To solve this, we need to find two consecutive whole numbers between which the square root of 104 falls. Since we know that \(10^2 = 100\) and \(11^2 = 121\), the square root of 104 must be greater than 10 but less than 11 because 104 falls between 100 and 121. Therefore, the correct numbers to place in the boxes are 10 and 11. The completed inequality should read "10 < √104 < 11".
Solving Inequality with Square Root
The image shows an inequality with two blank boxes and a square root sign in between. The inequality is structured as a number less than the square root of 86, which is then less than another number. To solve for the inequality, you first need to calculate the square root of 86 to determine what numbers the blanks could represent: √86 ≈ 9.27 (approximately, as the square root of 86 cannot be simplified to a simple integer) The numbers in the blank boxes should be close to but not equal to the square root of 86, with the left number being less than 9.27 and the right number being greater than 9.27. One potential set of numbers that would satisfy these conditions could be: Left box: 9 Right box: 10 Since 9 < √86 < 10 holds true with √86 being approximately 9.27.
Solving Absolute Value Inequality
The given inequality is \( 10 > |q - 10| \). This inequality states that the absolute value of \( q - 10 \) is less than 10. To solve this inequality, you will split it into two separate inequalities, one for each case of the absolute value function (positive and negative), and solve for \( q \). 1. For the positive case: \[ q - 10 < 10 \] Adding 10 to both sides gives you: \[ q < 20 \] 2. For the negative case: \[ -(q - 10) < 10 \] Simplifying further: \[ -q + 10 < 10 \] Subtracting 10 from both sides gives you: \[ -q < 0 \] Now, multiply both sides by -1 (remember that multiplying by -1 reverses the inequality): \[ q > 0 \] Combining both cases, the solution for \( q \) is the range where both conditions are met: \[ 0 < q < 20 \] So the set of all real numbers \( q \) that satisfies the original inequality is any number greater than 0 and less than 20.
Solving Inequality Involving Absolute Value
This inequality involves an absolute value. To solve it, you need to consider two cases because the absolute value of an expression represents its distance from zero on the number line, which could be either positive or negative. The inequality is \(10 > |q - 10|\). Case 1: \(q - 10\) is positive, which means that \(q\) is greater than 10. \[q - 10 < 10\] \[q < 20\] Case 2: \(q - 10\) is negative, which means that \(q\) is less than 10. When you remove the absolute value, you must also flip the inequality sign. \[ -(q - 10) < 10\] \[-q + 10 < 10\] \[-q < 0\] \[q > 0\] So, \(q\) must be greater than 0 and less than 20. Putting both cases together, you get the solution: \[0 < q < 20\] The values of \(q\) that satisfy the inequality are all the numbers between 0 and 20, not including 0 and 20 themselves.
Solving Absolute Value Inequality
The inequality in the image is written as `8|y| ≤ 24`. Here, we want to solve for the variable `y`. The term `|y|` represents the absolute value of `y`, which means it is the distance of `y` from zero on the number line and is always nonnegative. To solve the inequality for `y`, follow these steps: 1. Divide both sides of the inequality by 8 to isolate the absolute value expression: \( |y| ≤ 24 / 8 \) \( |y| ≤ 3 \) 2. The solution to the absolute value inequality \( |y| ≤ 3 \) means that `y` can be any number within the interval [-3, 3], including -3 and 3. In interval notation, the solution is written as: \( y ∈ [-3, 3] \) This interval represents all values of `y` that satisfy the original inequality.
Solving Absolute Value Inequality and Writing as a Compound Inequality
To solve the inequality \( 4|v - 3| \leq 28 \) and write it as a compound inequality, follow these steps: 1. First, isolate the absolute value on one side by dividing both sides of the inequality by 4: \[ |v - 3| \leq \frac{28}{4} \] \[ |v - 3| \leq 7 \] 2. Now, remember that the absolute value inequality \( |x| \leq a \) can be written as the compound inequality \( -a \leq x \leq a \). Applying this to your inequality, you get: \[ -7 \leq v - 3 \leq 7 \] 3. Adding 3 to all parts of the compound inequality to solve for \( v \) gives us: \[ -7 + 3 \leq v - 3 + 3 \leq 7 + 3 \] \[ -4 \leq v \leq 10 \] So, the solution to the inequality \( 4|v - 3| \leq 28 \) expressed as a compound inequality is \( -4 \leq v \leq 10 \).
Solving Absolute Value Inequality
The inequality provided in the image is: |u + 6| ≥ 46 To solve this absolute value inequality, we need to consider two separate cases because absolute value expressions represent the distance from zero and can be either positive or negative. Case 1: u + 6 is nonnegative If u + 6 is nonnegative, then |u + 6| just equals u + 6. So we simply remove the absolute value and solve the inequality: u + 6 ≥ 46 u ≥ 46 - 6 u ≥ 40 Case 2: u + 6 is negative If u + 6 is negative, then |u + 6| equals -(u + 6). In this case, when we remove the absolute value, we need to reverse the inequality sign: -(u + 6) ≥ 46 -u - 6 ≥ 46 -u ≥ 46 + 6 -u ≥ 52 Now, we multiply both sides by -1 and remember to reverse the inequality: u ≤ -52 Combining both cases, the solution to the inequality |u + 6| ≥ 46 is: u ≤ -52 or u ≥ 40 This means that any value of u that is less than or equal to -52 or greater than or equal to 40 satisfies the given inequality.
Solving Absolute Value Inequality
To solve the inequality \( 3 \left| \frac{x+3}{2} \right| \leq 9 \), we can proceed as follows: 1. Divide both sides of the inequality by 3 to isolate the absolute value expression: \[ \left| \frac{x+3}{2} \right| \leq \frac{9}{3} \] \[ \left| \frac{x+3}{2} \right| \leq 3 \] 2. The absolute value inequality \( |A| \leq B \) where \( B \) is non-negative (B ≥ 0) can be split into two inequalities: \[ -B \leq A \leq B \] Applying this concept, we get: \[ -3 \leq \frac{x+3}{2} \leq 3 \] 3. Multiply all parts of the compound inequality by 2 to get rid of the fraction: \[ -6 \leq x+3 \leq 6 \] 4. Now subtract 3 from all parts of the compound inequality to solve for \( x \): \[ -6 - 3 \leq x+3 - 3 \leq 6 - 3 \] \[ -9 \leq x \leq 3 \] So the solution set for the inequality is \( x \in [-9, 3] \).
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