Example Question - inequality representation
Here are examples of questions we've helped users solve.
Graphical Representation of a Linear Inequality Solution
<p>Simplify the given inequality \( (3x + 1)^2 \geq (3x + 1)(3) \)</p> <p>Expand both sides: \( 9x^2 + 6x + 1 \geq 9x + 3 \)</p> <p>Rearrange terms to set the inequality to zero: \( 9x^2 + 6x + 1 - 9x - 3 \geq 0 \)</p> <p>Simplify: \( 9x^2 - 3x - 2 \geq 0 \)</p> <p>Factor the quadratic inequality: \( (3x + 1)(3x - 2) \geq 0 \)</p> <p>Identify critical points where the inequality can change signs by setting each factor equal to zero: \( 3x + 1 = 0 \) and \( 3x - 2 = 0 \)</p> <p>Solve for x to find the critical points: \( x = -\frac{1}{3} \) and \( x = \frac{2}{3} \)</p> <p>Test intervals to determine where the inequality holds; use test points \( x < -\frac{1}{3} \), \( -\frac{1}{3} < x < \frac{2}{3} \), and \( x > \frac{2}{3} \)</p> <p>Choose test points such as \( x = -1 \), \( x = 0 \), and \( x = 1 \); verify that the inequality is satisfied for \( x < -\frac{1}{3} \) and \( x > \frac{2}{3} \) but not in the middle</p> <p>The solution is \( x \in (-\infty, -\frac{1}{3}] \) union \( [\frac{2}{3}, \infty) \)</p> <p>Therefore, the correct graph is D.</p>
Number Line Inequality Representation
The image shows a number line with a shaded region that starts at the number 8 and extends to the left, which indicates all numbers less than 8. The dot at number 8 is not filled in, which means that 8 itself is not included in the solution set. Therefore, the inequality that represents this graph is: x < 8 So the correct answer is A. x < 8.
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