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Example Question - inequality solution

Here are examples of questions we've helped users solve.

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 Linear Inequalities by Isolating Variables

To solve the inequality 5q + 3 < 37, you need to isolate the variable q. Here are the steps: 1. Subtract 3 from both sides: 5q + 3 - 3 < 37 - 3 2. This simplifies to: 5q < 34 3. Now, divide both sides by 5: 5q / 5 < 34 / 5 4. You get: q < 6.8 Now, let's look at the provided options: - \( q = -8 \) is less than 6.8, so it is a solution. - \( q = -5 \) is also less than 6.8, so it is a solution. - \( q = -2 \) is less than 6.8, so it is a solution. - \( q = -1 \) is less than 6.8, so it is also a solution. All four options are solutions to the inequality because they are all less than 6.8. However, if the question intends to ask for only one solution, they typically expect the greatest value that satisfies the inequality. Since \( q = -1 \) is the greatest value among the choices and still satisfies the inequality \( q < 6.8 \), it would be considered the best solution from the provided options.

Solving Absolute Value Inequalities with Two Cases

To solve the inequality \(7 \leq |7y - 9|\), we must consider two cases because the absolute value function \(|x|\) is defined as \(x\) when \(x \geq 0\) and \(-x\) when \(x < 0\). **Case 1**: \(7y - 9 \geq 0\) If \(7y - 9\) is non-negative, the inequality is simply: \(7 \leq 7y - 9\) Now, we solve for \(y\): \(7 + 9 \leq 7y\) \(16 \leq 7y\) Divide both sides by 7: \(y \geq \frac{16}{7}\) **Case 2**: \(7y - 9 < 0\) If \(7y - 9\) is negative, then \(|7y - 9| = -(7y - 9)\), so the inequality becomes: \(7 \leq -(7y - 9)\) Now, we solve for \(y\): \(7 \leq -7y + 9\) Rearrange: \(7y \leq 9 - 7\) \(7y \leq 2\) Divide both sides by 7: \(y \leq \frac{2}{7}\) Combining the results from both cases gives us the complete solution set for the inequality: \(\frac{2}{7} \geq y \geq \frac{16}{7}\) However, if you look carefully at the inequalities derived from both cases, you'll notice that there's actually no overlap between the two solution sets, since it's not possible for a number \(y\) to be simultaneously greater than or equal to approximately \(2.29\) and less than or equal to approximately \(0.29\). This suggests we need to correct the combined inequality to reflect the actual solution. The correct combined solution for the inequality considering both cases would be: \(y \leq \frac{2}{7}\) or \(y \geq \frac{16}{7}\) Thus, the solution set for the inequality \(7 \leq |7y - 9|\) is that \(y\) can be in the interval \((-\infty, \frac{2}{7}]\) or \([\frac{16}{7}, +\infty)\).

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 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 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 a System of Inequalities

我们要解决的问题是找到一个不等式，它能同时满足\( x + 2 < 9 \)和\( x - 1 < 3 \)。首先，我们解第一个不等式： \( x + 2 < 9 \) \( x < 9 - 2 \) \( x < 7 \) 接着解第二个不等式： \( x - 1 < 3 \) \( x < 3 + 1 \) \( x < 4 \) 因为x必须同时满足这两个条件，所以x的实际取值范围在更严格的条件下，也就是\( x < 4 \)。所以我们得出结论，可以满足这两个不等式的x的区间是： \( x < 4 \) 这个不等式代表的是所有满足两个给定不等式条件的x值。

Solving an Inequality

Sure, I'll help solve the inequality given in the image: \((-3x + 7) + (-4x - 8) \geq -3x + 7\) First, let's simplify the left side of the inequality by combining like terms: \((-3x - 4x) + (7 - 8)\) This simplifies to: \(-7x - 1\) Now, substitute this back into the inequality: \(-7x - 1 \geq -3x + 7\) To solve for \(x\), let's move all the terms with \(x\) on one side and the constant terms on the other side. First, add \(7x\) to both sides of the inequality: \(-7x - 1 + 7x \geq -3x + 7 + 7x\) This simplifies to: \(-1 \geq 4x + 7\) Next, subtract \(7\) from both sides: \(-1 - 7 \geq 4x + 7 - 7\) Which is: \(-8 \geq 4x\) Now, divide both sides by \(4\) to solve for \(x\): \(-2 \geq x\), or equivalently written as \(x \leq -2\) So the solution to the inequality is \(x \leq -2\).