Example Question - absolute value inequality
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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 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
To solve the inequality \( 9 - 8|r + 5| > -11 \), we need to isolate the absolute value expression on one side. First, let's move the constant term on the left to the other side by subtracting 9 from both sides: \( 9 - 8|r + 5| - 9 > -11 - 9 \) \( -8|r + 5| > -20 \) Next, to get \( |r + 5| \) by itself, divide both sides of the inequality by -8. Remember that dividing by a negative number flips the direction of the inequality: \( \frac{-8|r + 5|}{-8} < \frac{-20}{-8} \) \( |r + 5| < \frac{20}{8} \) \( |r + 5| < \frac{5}{2} \) or \( |r + 5| < 2.5 \) The absolute value inequality \( |r + 5| < 2.5 \) means that \( r + 5 \) must be less than 2.5 and greater than -2.5 (because the distance from zero is less than 2.5). So we can break this into two separate inequalities: \( r + 5 < 2.5 \) \( r + 5 > -2.5 \) Let's solve each of these: For \( r + 5 < 2.5 \): Subtract 5 from both sides: \( r < 2.5 - 5 \) \( r < -2.5 \) For \( r + 5 > -2.5 \): Subtract 5 from both sides: \( r > -2.5 - 5 \) \( r > -7.5 \) Combining both inequalities, we get the solution for \( r \): \( -7.5 < r < -2.5 \) This is the set of all real numbers greater than -7.5 and less than -2.5.
Solving Absolute Value Inequalities
To solve the inequality -9|c| < -50, divide both sides of the inequality by -9. However, remember that when you divide or multiply an inequality by a negative number, the direction of the inequality sign must be reversed. Dividing both sides by -9, we get: |c| > 50 / 9 The absolute value inequality |c| > 50/9 indicates that c can be greater than 50/9 or less than -50/9. Hence, the solution to the inequality is: c < -50/9 or c > 50/9 Since 50/9 is approximately 5.56, the solution in decimal form is: c < -5.56 or c > 5.56
Solving Inequalities Involving Absolute Value
To solve the inequality \(-9|c| < -50\), let's start by isolating the absolute value term. Firstly, divide both sides by -9 to get the absolute value by itself. It's important to remember that when we divide or multiply both sides of an inequality by a negative number, the direction of the inequality symbol changes. So the inequality will reverse in this case. \[ -9|c| < -50 \quad / :(-9) \] \[ |c| > \frac{-50}{-9} \] \[ |c| > \frac{50}{9} \] Since the absolute value of a number is always nonnegative, we can write this as two separate inequalities: \[ c > \frac{50}{9} \quad or \quad c < -\frac{50}{9} \] So the solution to the inequality is that \(c\) is either greater than \(\frac{50}{9}\) or less than \(-\frac{50}{9}\).
Solving an Inequality Involving Absolute Value
To solve the given inequality, \( 8 - 7|-6s| < -6 \), let's first simplify the expression inside the absolute value sign by multiplying 7 and the absolute value of \(-6s\): \[ 8 - 7 \cdot |-6s| < -6 \] Let \( A = |-6s| \), then we have: \[ 8 - 7A < -6 \] Now, let's solve for \( A \) by isolating it on one side: \[ -7A < -6 - 8 \] \[ -7A < -14 \] Divide both sides by -7, and remember to reverse the inequality sign since we are dividing by a negative number: \[ A > 2 \] But \( A \) was defined as \( |-6s| \), so we substitute back in: \[ |-6s| > 2 \] Now we have to consider the two cases due to the absolute value. This inequality splits into two separate inequalities: 1. When the expression inside the absolute value is positive or zero: \[ -6s > 2 \] \[ s < -\frac{1}{3} \] 2. When the expression inside the absolute value is negative: \[ -6s < -2 \] \[ s > \frac{1}{3} \] Since no values of \( s \) can be simultaneously less than \(-\frac{1}{3}\) and greater than \(\frac{1}{3}\), this inequality has no solution. It means there are no values of \( s \) that can satisfy the original inequality \( 8 - 7|-6s| < -6 \).
Solving Absolute Value Inequality
The equation provided in the image given is an absolute value inequality involving the variable \( p \): \[ |2p| - 8 \geq 13 \] To solve this inequality, we'll perform the following steps: 1. Isolate the absolute value expression on one side of the inequality. \[ |2p| \geq 13 + 8 \] \[ |2p| \geq 21 \] 2. Look at both scenarios which can make the absolute value expression true. Since the absolute value of an expression is the distance of that expression from zero, it can be either positive or negative. Therefore: a) \( 2p \geq 21 \) b) \( 2p \leq -21 \) 3. Solve both inequalities for \( p \): a) For \( 2p \geq 21 \): \[ p \geq \frac{21}{2} \] \[ p \geq 10.5 \] b) For \( 2p \leq -21 \): \[ p \leq \frac{-21}{2} \] \[ p \leq -10.5 \] So the solution set for the inequality \( |2p| - 8 \geq 13 \) is \( p \geq 10.5 \) or \( p \leq -10.5 \). This means that \( p \) must be less than or equal to -10.5 or greater than or equal to 10.5.
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
The inequality provided in the image is \( 8|y| \leq 24 \). To solve for \( y \), we need to isolate \( y \). We start by dividing both sides of the inequality by 8: \( |y| \leq 24 / 8 \\ |y| \leq 3 \) The absolute value sign indicates that whatever value \( y \) takes, when we take its absolute value, it should be less than or equal to 3. This means that \( y \) can be any number in the range of -3 to 3 inclusive. Thus, the solution set for \( y \) is: \( -3 \leq y \leq 3 \)
Absolute Value Inequality Solution
The inequality is \( 3|d| + 5 < 47 \). First, we'll isolate the absolute value expression by subtracting 5 from both sides of the inequality: \( 3|d| + 5 - 5 < 47 - 5 \) \( 3|d| < 42 \) Now, divide both sides by 3 to solve for the absolute value of d: \( \frac{3|d|}{3} < \frac{42}{3} \) \( |d| < 14 \) Since we have an absolute value inequality, we know that \( |d| < 14 \) means that d is less than 14 and greater than -14. Therefore, the solution set for d is: \( -14 < d < 14 \)
Solving Absolute Value Inequality
To solve the inequality \( 3|1 - 9b| \geq 28 \), you can follow these steps: 1. Divide both sides of the inequality by 3 to isolate the absolute value expression: \( |1 - 9b| \geq \frac{28}{3} \) 2. Recognize that an absolute value expression \( |x| \geq a \) where \( a > 0 \) can be split into two separate inequalities: \( x \geq a \) or \( x \leq -a \) In this case, \( x \) corresponds to \( 1 - 9b \) and \( a \) corresponds to \( \frac{28}{3} \). 3. Set up the two inequalities: \( 1 - 9b \geq \frac{28}{3} \) and \( 1 - 9b \leq -\frac{28}{3} \) 4. Solve each inequality separately: For \( 1 - 9b \geq \frac{28}{3} \): a) Subtract 1 from both sides: \( -9b \geq \frac{28}{3} - 1 \) \( -9b \geq \frac{28}{3} - \frac{3}{3} \) \( -9b \geq \frac{25}{3} \) b) Divide by -9, remembering to flip the inequality sign when dividing by a negative: \( b \leq -\frac{25}{27} \) For \( 1 - 9b \leq -\frac{28}{3} \): a) Subtract 1 from both sides: \( -9b \leq -\frac{28}{3} - 1 \) \( -9b \leq -\frac{28}{3} - \frac{3}{3} \) \( -9b \leq -\frac{31}{3} \) b) Divide by -9, flipping the inequality sign: \( b \geq \frac{31}{27} \) 5. Combine the two parts of the solution: \( b \leq -\frac{25}{27} \) or \( b \geq \frac{31}{27} \) This is the solution to the 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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