Example Question - solving inequalities
Here are examples of questions we've helped users solve.
Solving a Fractional Inequality Problem
<p>The provided inequality is: </p> <p>\(\frac{4}{x-2} > \frac{7}{x-3}\)</p> <p>To solve it, first find a common denominator: </p> <p>\((x-2)(x-3)\)</p> <p>Then multiply both sides by the common denominator, being careful to consider that it can change sign based on the value of \(x\): </p> <p>\(4(x-3) > 7(x-2)\)</p> <p>Now distribute and move all terms to one side: </p> <p>\(4x - 12 > 7x - 14\)</p> <p>Subtract \(4x\) and add \(14\) to both sides: </p> <p>\(-12 + 14 > 7x - 4x\)</p> <p>Simplify: </p> <p>\(2 > 3x\)</p> <p>Now, divide by 3: </p> <p>\(\frac{2}{3} > x\)</p> <p>So, the solution in terms of \(x\) is: </p> <p>\(x < \frac{2}{3}\)</p> <p>However, since the denominators in the original fractions were \(x-2\) and \(x-3\), we must exclude 2 and 3 from the domain of \(x\). Thus, along with the inequality above, we have the domain restrictions: </p> <p>\(x \neq 2\) and \(x \neq 3\)</p> <p>Combining these, the final solution is: </p> <p>\(x < \frac{2}{3}\), \(x \neq 2\), \(x \neq 3\)</p>
Solving Inequalities with Variable g
The inequality given in the image is: 5g + 3 ≤ 37 To solve for g, we follow these steps: 1. Subtract 3 from both sides to isolate the term with g on one side: 5g + 3 - 3 ≤ 37 - 3 Which simplifies to: 5g ≤ 34 2. Now, divide both sides by 5 to solve for g: \( \frac{5g}{5} \) ≤ \( \frac{34}{5} \) This gives us: g ≤ 6.8 So, any value of g that is less than or equal to 6.8 will satisfy the inequality. Looking at the options provided: - g = 8: this does not satisfy the inequality because 8 is greater than 6.8. - g = 5: this does satisfy the inequality because 5 is less than 6.8. - g = -2: this also satisfies the inequality because -2 is less than 6.8. - g = -1: this too satisfies the inequality because -1 is less than 6.8. Since three of the options satisfy the inequality and we are asked for a solution to the inequality, we have to identify which one of the options is correct. Based on the information provided in the image, it seems we are asked for a single solution, but there are three possible solutions. In this scenario, without further information, it would be best to state that g = 5, g = -2, and g = -1 all are solutions to the given inequality since they all result in a true statement when substituted back into the inequality.
Solving Inequalities with Variable Isolation
To solve the inequality \(5g + 3 \leq 37\), we need to isolate the variable \(g\). Here are the steps: 1. Subtract 3 from both sides: \[5g + 3 - 3 \leq 37 - 3\] \[5g \leq 34\] 2. Divide both sides by 5: \[\frac{5g}{5} \leq \frac{34}{5}\] \[g \leq \frac{34}{5}\] \[g \leq 6.8\] Now, let's look at the answer choices to find which ones are less than or equal to 6.8: - \(g = -8\) is less than 6.8; hence it's a valid solution. - \(g = -5\) is less than 6.8; hence it's a valid solution. - \(g = -2\) is less than 6.8; hence it's a valid solution. - \(g = -1\) is less than 6.8; hence it's a valid solution. All of the given choices \(g = -8, g = -5, g = -2, g = -1\) are solutions to the inequality \(5g + 3 \leq 37\).
Solving Inequalities with Variable q
Let's solve the inequality given in the image: 5q + 3 < 37 Our first step is to isolate the term with the variable q. To do that, we subtract 3 from both sides of the inequality. 5q + 3 - 3 < 37 - 3 This simplifies to: 5q < 34 Next, we divide both sides by 5 to solve for q: 5q / 5 < 34 / 5 q < 6.8 The value of q must be less than 6.8. Now let's check the provided options to see which one is a solution to the inequality: a) q = 8 Since 8 is greater than 6.8, this is not a solution. b) q = 5 Since 5 is less than 6.8, this is a solution. c) q = 2 Since 2 is less than 6.8, this is also a solution. d) q = -1 Since -1 is less than 6.8, this is also a solution. The question asks for a solution to the inequality. The correct answers are the ones where q is less than 6.8. Therefore, the solutions from the given options are q = -1, q = 2, and q = 5. However, since only one option can be chosen and all three are correct, without additional context or instructions, it is impossible to determine which single option the question intends as the answer. However, if the question requires to choose all possible solutions from the options given, then q = 5, q = 2, and q = -1 would all be acceptable solutions.
Solving Equations and Inequalities in Mathematics
Pour résoudre le problème présenté dans l'image, commençons par le premier point : 1. Résoudre dans R l'équation : \[ \frac{1}{3}x + 3 = 0 \] \[ \frac{2x + 3}{x + 1} = \frac{(x + 1)(x - 3)}{x + 1} \] Résolvons d'abord l’équation simple \( \frac{1}{3}x + 3 = 0 \): Multiplications des deux côtés par 3 pour se débarrasser du facteur \( \frac{1}{3} \): \[ x + 9 = 0 \] Substrayons 9 des deux côtés de l’équation : \[ x = -9 \] Le côté droit de l’équation a une expression fractionnaire que nous pouvons simplifier en annulant les termes communs (x + 1) en supposant que x n'est pas égal à -1 (car cela annulerait le dénominateur, ce qui n'est pas permis). Cela nous donne directement (x - 3). Ensuite, vérifions si les deux côtés de l’équation sont égaux pour x = -9 : À gauche nous avons \( \frac{2(-9) + 3}{-9 + 1} = \frac{-18 + 3}{-8} = \frac{-15}{-8} = \frac{15}{8} \) À droite nous avons (-9 + 1)(-9 - 3) = (-8)(-12) = 96 Puisque \( \frac{15}{8} \) n'est pas égal à 96, l'équation n'est pas vraie pour x = -9. Donc, x = -9 n’est pas une solution pour l’équation donnée. Passons au deuxième point de l'exercice : 2. Soit \( p(x) = (x - 2)(x + 5) \) a) Résoudre dans R l'équation p(x) = 0 Pour résoudre cette équation, nous cherchons les valeurs de x pour lesquelles le produit est nul. Un produit de facteurs est nul si au moins l'un des facteurs est nul. Donc, \( x - 2 = 0 \) ou \( x + 5 = 0 \) Cela donne deux solutions : \[ x = 2 \] et \[ x = -5 \] b) Résoudre dans R l'inéquation p(x) > 0 Pour résoudre cette inéquation, nous utilisons les solutions de l'équation trouvées précédemment pour déterminer les intervalles sur lesquels le produit est positif. Les racines de l'équation divisent la ligne numérique en intervalles sur lesquels le signe du produit ne change pas. On a : - Pour \( x < -5 \), le produit (x - 2)(x + 5) est positif car on a un nombre négatif multiplié par un autre nombre négatif. - Pour \( -5 < x < 2 \), le produit est négatif car on a un nombre positif multiplié par un nombre négatif. - Pour \( x > 2 \), le produit est à nouveau positif car on a deux nombres positifs. Donc, les intervalles sur lesquels p(x) > 0 sont : \[ x \in ]-\infty; -5[ \cup ]2; +\infty[ \] Pour le troisième point : 3. Considérons l'équation (E) : x² - 2x + 1 = 0 Nous savons déjà que \( x_1 = \sqrt{2} - 1 \) est une solution de l'équation (E). Pour trouver la deuxième solution, remarquons que l'équation est une équation du second degré qui peut être écrite sous forme factorisée car c'est une équation du type \( (x - a)^2 = 0 \), où \( a \) est le terme racine. L'équation donnée est la forme développée de \( (x - 1)^2 \): \[ (x - 1)^2 = x^2 - 2x + 1 \] Pour que cette équation soit nulle, il est nécessaire que \( x - 1 = 0 \), d'où \( x = 1 \). Ceci est la deuxième solution de l'équation. Pour le quatrième point, l'inéquation à résoudre est : 4. Résoudre dans R l'inéquation \( 4x - 3\sqrt{x} + 1 < 0 \) Pour trouver les valeurs de x qui satisfont cette inéquation, il serait utile de factoriser l'expression ou d'utiliser une autre technique appropriée. Dans ce cas, la résolution exacte pourrait devenir complexe selon le niveau d'études attendu. Si vous avez besoin d'une méthode spécifique pour résoudre cette inéquation, veuillez préciser et je pourrai vous guider en conséquence.
Solving Inequalities with Square Roots
The image seems to show an inequality with a square root, configured as two blank boxes with "< √130 <" in between. The task appears to be to find two numbers that satisfy the inequality where the left number is smaller than the square root of 130 and the right number is larger than the square root of 130. To solve for the inequality, we need to calculate the square root of 130. Since √130 is not a perfect square, we'll get a decimal answer. Using a calculator or an appropriate method, we find that √130 is approximately 11.4018. Therefore, we need two numbers such that the one on the left is less than 11.4018 and the one on the right is more than 11.4018. Two such integers could be 11 and 12, respectively. However, any two numbers that fit the criteria -- for example, 11.3 (left) and 11.5 (right) -- would be a correct answer to fill in the boxes. In the given inequality, the blank boxes could be filled in as follows for an integer solution: 11 < √130 < 12
Solving Inequalities with Negative Square Roots
The image displays an inequality involving the negative square root of 122, which is written as "-√122". To solve the inequality, we will first find the value of √122. √122 ≈ 11.045 (rounded to three decimal places) Since we need the negative square root, we take the negative of this value: -√122 ≈ -11.045 Now, if the inequality signs in the image are meant to encapsulate values around -√122, we would be looking for numbers that are slightly less than and slightly greater than -11.045. Thus, any number less than -11.045 will fulfill the left part of the inequality, and any number greater than -11.045 will fulfill the right part of the inequality. To be clear, without the specific values or context for what numbers should be on either side of the inequality signs, we can't provide an exact numerical range. The inequality simply indicates numbers less than and greater than approximately -11.045. If the problem is asking for integers, the integers immediately less than and greater than -11.045 are -12 and -11, respectively.
Inequality Statement with Square Root of 68
The task here appears to be to fill in the blanks with numbers to make the inequality statement true, with the middle term being the square root of 68. To solve this, we first find the approximate value of the square root of 68. We know that: - √64 = 8 - √81 = 9 Since 68 is between 64 and 81, √68 will be between 8 and 9. Now that we know √68 is between 8 and 9, we can fill in the blanks with numbers to make the statement true. A logical approach would be to use the numbers 8 and 9 since the square root of 68 is between them. The complete statement should read: 8 < √68 < 9 This means that the square root of 68 is greater than 8 and less than 9, which is a true statement.
Solving Absolute Value Inequalities
The inequality presented here is \(7 \leq |7y - 9|\). To solve this, we need to consider the absolute value |7y - 9| and split the inequality into two cases because the expression inside the absolute value can be either positive or negative. The absolute value inequality says that the expression inside the absolute value is either greater than or equal to 7 or less than or equal to -7. First, we'll handle the case where the expression inside the absolute value is non-negative: 1. \(7y - 9 \geq 7\) 2. \(7y \geq 16\) 3. \(y \geq \frac{16}{7}\) Next, we handle the case where the expression inside the absolute value is non-positive: 1. \(7y - 9 \leq -7\) 2. \(7y \leq 2\) 3. \(y \leq \frac{2}{7}\) Combining both inequalities, we get the solution set for y: \(y \leq \frac{2}{7}\) or \(y \geq \frac{16}{7}\). Graphically, this means y is either in the interval \((-\infty, \frac{2}{7}]\) or in the interval \([\frac{16}{7}, \infty)\).
Solving Inequalities with Absolute Value
To solve the inequality \( |t - 75| \leq 15 \), we need to split it into two separate inequalities, because the absolute value of a number is the distance from zero on the number line, and it can be expressed as the value being either greater than or equal to 0 or less than or equal to 0. The two cases for \( |t - 75| \leq 15 \) are: 1. \( t - 75 \leq 15 \) 2. \( -(t - 75) \leq 15 \) or equivalently \( t - 75 \geq -15 \) For the first case (1): \( t - 75 \leq 15 \) Adding 75 to both sides of the inequality gives us: \( t \leq 90 \) For the second case (2): \( t - 75 \geq -15 \) Adding 75 to both sides of this inequality gives us: \( t \geq 60 \) So, the solution to \( |t - 75| \leq 15 \) is the set of values of \( t \) that satisfy both conditions: \( 60 \leq t \leq 90 \) This means that the value of \( t \) is between 60 and 90, inclusive.
Solving Absolute Value Inequalities with a Non-Negative Constant
This inequality involves an absolute value. The statement |x| ≥ k, where k is a non-negative number, is equivalent to saying that x ≤ -k or x ≥ k, since the absolute value indicates the distance of a number from zero, not the direction. Given the inequality: 50 ≤ |1 - g + 950| we can split this into two separate inequalities because the expression inside the absolute value must either be greater than or equal to 50 or less than or equal to -50. Therefore, we have: 1. 1 - g + 950 ≥ 50 2. 1 - g + 950 ≤ -50 Let's solve each inequality. For the first inequality: 1 - g + 950 ≥ 50 Subtract 950 from both sides: 1 - g ≥ 50 - 950 1 - g ≥ -900 Now, add g to both sides: 1 ≥ g - 900 Finally, add 900 to both sides: g ≤ 901 For the second inequality: 1 - g + 950 ≤ -50 Subtract 950 from both sides: 1 - g ≤ -50 - 950 1 - g ≤ -1000 Add g to both sides: 1 + g ≤ -1000 Add 1 to both sides: g ≤ -1001 Combining both sets of g values, we have: g ≤ 901 and g ≤ -1001 However, since g cannot be simultaneously less than or equal to both 901 and -1001, we consider the entire range of g that satisfies either condition. Thus, the solution to the inequality is: g ≤ 901 Since -1001 is much less than 901, the latter encompasses all possible values of g that could satisfy the inequality.
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 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.
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