How do I solve the inequality 3x – 2 ≤ 7?To solve the inequality 3x – 2 ≤ 7, first add 2 to both sides to get 3x ≤ 9. Then, divide both sides by 3 to isolate x, resulting in x ≤ 3. Therefore, the solution to the inequality is x ≤ 3.
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How do you solve for x in a linear equation like 2x + 5 = 15?
How do you solve for x in a linear equation like 2x + 5 = 15?To solve the linear equation 2x + 5 = 15, first subtract 5 from both sides to get 2x = 10. Then, divide both sides by 2 to isolate x, resulting in x = 5.
How do you use the Pythagorean theorem to find the unknown side length of a right triangle in a real-world problem?
How do you use the Pythagorean theorem to find the unknown side length of a right triangle in a real-world problem?To use the Pythagorean theorem in a real-world problem, identify the right triangle’s sides: the two legs (a and b) and the hypotenuse (c). Apply the formula a² + b² = c². Solve for the unknown side by rearranging the equation and taking the square root if necessary. For example, in construction, you can determine the length of a ladder needed to reach a certain height by knowing the distance from the wall.
How do you find the exact values of trigonometric functions for angles that aren’t standard, like 75 degrees or 15 degrees?
How do you find the exact values of trigonometric functions for angles that aren’t standard, like 75 degrees or 15 degrees?To find the exact values of trigonometric functions for non-standard angles like 75° or 15°, use angle sum and difference identities. For example, sin(75°) = sin(45° + 30°) = sin(45°)cos(30°) + cos(45°)sin(30°). Similarly, cos(75°) and tan(75°) can be calculated using the same identities.
If a train travels at a constant speed of 75 miles per hour, how long will it take for the train to travel 262.5 miles? Additionally, if the train continues traveling at the same speed, how far will it travel in 7 hours?
If a train travels at a constant speed of 75 miles per hour, how long will it take for the train to travel 262.5 miles? Additionally, if the train continues traveling at the same speed, how far will it travel in 7 hours?To determine the time taken to travel 262.5 miles at 75 miles per hour, divide the distance by the speed: 262.5 miles ÷ 75 miles per hour = 3.5 hours. To find the distance traveled in 7 hours at the same speed, multiply the speed by the time: 75 miles per hour × 7 hours = 525 miles.
How do you solve systems of linear equations using the substitution method?
How do you solve systems of linear equations using the substitution method?The substitution method for solving systems of linear equations involves isolating one variable in one equation and substituting this expression into the other equation. This reduces the system to a single equation with one variable, which can then be solved. Finally, the value is substituted back into the original equation to find the other variable.
What is the difference between a simple random sample and a stratified random sample, and in what situations might you use each method?
What is the difference between a simple random sample and a stratified random sample, and in what situations might you use each method?A simple random sample (SRS) involves selecting individuals from a population entirely by chance, ensuring each individual has an equal probability of being chosen. In contrast, a stratified random sample (SRS) divides the population into distinct subgroups (strata) and then randomly samples from each subgroup. Use SRS for homogeneous populations, and stratified sampling for heterogeneous populations to ensure representation of all subgroups.
How can regression analysis be applied to predict future trends in a dataset, and what are the potential pitfalls in dealing with model overfitting using multiple predictors?
How can regression analysis be applied to predict future trends in a dataset, and what are the potential pitfalls in dealing with model overfitting using multiple predictors?Regression analysis predicts future trends by modeling relationships between variables. It uses historical data to fit a model, which can then forecast future values. However, using multiple predictors can lead to overfitting, where the model captures noise rather than the true underlying trend. This reduces predictive accuracy. Techniques like cross-validation, regularization, and simplifying the model can mitigate overfitting.
What is the Pythagorean Theorem and can you provide an example of how it is used?
What is the Pythagorean Theorem and can you provide an example of how it is used?The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Mathematically, it’s expressed as a² + b² = c². For example, if a triangle has sides of lengths 3 and 4, the hypotenuse would be 5, since 3² + 4² = 9 + 16 = 25, and √25 = 5.
How do you determine if results from an experiment with multiple treatment groups meet the assumptions needed for an ANOVA test to verify their significance?
How do you determine if results from an experiment with multiple treatment groups meet the assumptions needed for an ANOVA test to verify their significance?To determine if results from an experiment with multiple treatment groups meet the assumptions for an ANOVA test, check for normality, homogeneity of variances, and independence. Use tests like Shapiro-Wilk for normality, Levene’s test for equal variances, and ensure random sampling for independence.
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Determine the cosine of the angle t on the unit circle when the sine of t is 1/2
Answer 1 To find the cosine of the angle $t$ on the unit circle when the sine of $t$ is $\frac{1}{2}$, we can use the Pythagorean identity:\n $ \sin^2(t) + \cos^2(t) = 1 $\n Given that $\sin(t) = \frac{1}{2}$, we substitute and solve for $\cos(t)$:\n...
Find the angle θ in the unit circle where the sum of sin(θ) and cos(θ) equals 15
Answer 1 To find the angle $ \theta $ where the sum of $ \sin(\theta) $ and $ \cos(\theta) $ equals 1.5, we start with the equation: $ \sin(\theta) + \cos(\theta) = 1.5 $ We can use the Pythagorean identity: $ \sin^2(\theta) + \cos^2(\theta) = 1 $...
Find the sine and cosine values for an angle of pi/4 on the unit circle
Answer 1 To find the sine and cosine values for an angle of $ \frac{\pi}{4} $ on the unit circle, we can use the known values of the unit circle. For an angle of $ \frac{\pi}{4} $: $ \sin\left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} $ $...
Find the exact values of sin and cos for all solutions in the third quadrant for the equation 2*sin(x) + 3*cos(x) = 1
Answer 1 To find the exact values of $\sin(x)$ and $\cos(x)$ for all solutions in the third quadrant for the equation $2\sin(x) + 3\cos(x) = 1$, consider the trigonometric identity:$ \sin^2(x) + \cos^2(x) = 1 $In the third quadrant, both $ \sin(x) $...
Find the values of cos(θ) and sin(θ) at specific angles on the unit circle
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Find the value of tan(7π/6) and explain using the unit circle
Answer 1 To find the value of $ \tan(\frac{7\pi}{6}) $ using the unit circle:1. Locate the angle $\frac{7\pi}{6}$ on the unit circle. This angle is in the third quadrant.2. The reference angle for $\frac{7\pi}{6}$ is $\frac{\pi}{6}$.3. In the third...