Example Question - solution set
Here are examples of questions we've helped users solve.
Graphical Representation of a Rational Inequality Solution
<p>To find the solution to the inequality \((3x + 1)^2 > 3(3x + 1)\), we start by expanding and simplifying:</p> <p>\((3x + 1)(3x + 1) > 3(3x + 1)\)</p> <p>\(9x^2 + 6x + 1 > 9x + 3\)</p> <p>\(9x^2 + 6x - 9x + 1 - 3 > 0\)</p> <p>\(9x^2 - 3x - 2 > 0\)</p> <p>We then factor the quadratic expression, if possible, or use the quadratic formula to find the roots (the values of \(x\) where \(9x^2 - 3x - 2 = 0\)).</p> <p>Factoring gives us:</p> <p>\((3x + 1)(3x - 2) > 0\)</p> <p>The roots are \(x = -\frac{1}{3}\) and \(x = \frac{2}{3}\). The inequality will change signs at each of these points. To determine where the expression is positive, we can test intervals that are defined by these roots. We choose test points from intervals \((-∞, -\frac{1}{3})\), \((- \frac{1}{3}, \frac{2}{3})\), and \((\frac{2}{3}, ∞)\).</p> <p>Upon testing, we find the inequality is satisfied for \(x < -\frac{1}{3}\) and \(x > \frac{2}{3}\).</p> <p>On the number line, the solution will be shown with an open circle at \(x = -\frac{1}{3}\) and \(x = \frac{2}{3}\) and shading to the left of \(x = -\frac{1}{3}\) and to the right of \(x = \frac{2}{3}\), because these are not included in the solution set.</p> <p>Therefore, the correct answer is option D.</p>
Identifying the Range of an Inequality
<p>Verilen eşitsizlikteki sayılar arasındaki mesafeyi ve aralığı bulmamız gerekiyor.</p> <p>Bir sayı çizelgesi üzerinde, -5 ile 0 arasındaki her sayı bu eşitsizliği sağlamaktadır. </p> <p>Eşitsizlik \(-5 < x < 0\) şeklinde ifade edilebilir. </p> <p>Yani sayı çizgisi üzerindeki açık parantezler (-5) ve (0) arasındaki sayıları kapsar. </p>
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: \( -3 \leq y \leq 3 \)
Solving Absolute Value Inequalities
To solve the inequality \( |c| - 1 \geq 11 \), you need to isolate the absolute value on one side of the inequality. First, add 1 to both sides of the inequality: \[ |c| - 1 + 1 \geq 11 + 1 \] \[ |c| \geq 12 \] The absolute value \( |c| \) represents the distance from 0 on the number line, and it can be either positive or negative. Since \( |c| \geq 12 \), this means that c can either be greater than or equal to 12, or less than or equal to -12. Split the inequality into two cases: 1. \( c \geq 12 \) (when c is positive or zero) 2. \( c \leq -12 \) (when c is negative) Therefore, the solution set for \( |c| - 1 \geq 11 \) is \( c \geq 12 \) or \( c \leq -12 \).
Graphing an Inequality on a Number Line
The inequality given in the image is \( t \geq -3 \). To graph this inequality on a number line: 1. Locate the point -3 on the number line. 2. Since the inequality includes "greater than or equal to" (as indicated by the symbol \(\geq\)), you need a solid circle or dot at -3. This shows that -3 is part of the solution set. 3. Shade the number line to the right of -3, indicating all numbers greater than -3 are also included in the solution set. Now let's examine the provided options to see which graph corresponds to these instructions: A. This graph shows a number line with a solid dot at -3 and shading towards the left, which means values less than -3. This does not match the inequality. B. This graph shows a number line with an open circle at -3 (indicating that -3 is not included) and shading towards the right. This is not correct because the inequality specifies that -3 is included. C. This graph shows a number line with a solid dot at -3 and shading to the right, which seems to match the inequality \( t \geq -3 \). D. This graph depicts a number line with a solid dot at 3 and shading to the right; however, this does not correspond to the inequality provided. The correct answer is C, as it correctly represents the inequality \( t \geq -3 \).
Solving Absolute Value Inequalities
The equation given in the image is an inequality involving an absolute value: \[ \left| \frac{x + 3}{2} \right| \geq 4 \] To solve this inequality, we must consider two cases because the absolute value expression can be either positive or negative: **Case 1: The expression inside the absolute value is positive or zero** \[ \frac{x + 3}{2} \geq 4 \] To solve this, multiply both sides by 2 to get rid of the denominator: \[ x + 3 \geq 8 \] Now, subtract 3 from both sides: \[ x \geq 5 \] **Case 2: The expression inside the absolute value is negative** \[ -\left( \frac{x + 3}{2} \right) \geq 4 \] Again, multiply both sides by 2: \[ -(x + 3) \geq 8 \] Distribute the negative sign: \[ -x - 3 \geq 8 \] Now, add 3 to both sides: \[ -x \geq 11 \] Finally, multiply both sides by -1 and remember to flip the inequality sign when multiplying or dividing by a negative number: \[ x \leq -11 \] Combining both cases gives us the solution set: \[ x \leq -11 \quad \text{or} \quad x \geq 5 \] These are the values of x that satisfy the original inequality.
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