Solve the inequality 2x - 3 < x + 2 ≤ 3x + 5. Show your work.

Mathematics

1 answer:

grandymaker [24]3 years ago

5 0

Answer:

-3/2 ≤ x < 5

Step-by-step explanation:

Begin the solution of this inequality 2x - 3 < x + 2 ≤ 3x + 5 by combining the constants: Subtracting 2 from 2x - 3, x + 2 and 3x + 5 yields:

2x - 5 < x ≤ 3x + 3.

Focus on the first part of this inequality, 2x - 5 < x simplifies to x - 5 < 0 if we subtract x from both sides. Then x < 5 is part of the solution.

Focus now on the second part of this inequality: x ≤ 3x + 3. Subtracting x from both sides yields 0 ≤ 2x + 3, or -3 ≤ 2x, or 2x ≥ -3, or x ≥ -3/2.

Re-combining these two separate inequalities results in:

-3/2 ≤ x < 5 (equivalent to -1.5 ≤ x < 5).

Example: suppose we choose a test number from this interval. Does this test number satisfy the original equation? Choose x = 1.

Is 2(1) - 3 < 1 + 2 ≤ 3(1) + 5 true?

Is 2 - 3 < 3 ≤ 8 true? YES. So x = 1 is part of the solution set -3/2 ≤ x < 5.

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In a certain year, when she was a high school senior, Idonna scored 671 on the mathematics part of the SAT. The distribution of

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Answer:

Idonna's standardized score is 1.41.

Jonathan's standardized score is 0.55.

A.) Idonna's score is higher than Jonathan's

Step-by-step explanation:

Z-score:

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Idonna scored 671 on the mathematics part of the SAT. The distribution of SAT math scores in that year was Normal with mean 509 and standard deviation 115.

This means that her standardized score is Z when $X = 671, \mu = 509, \sigma = 115$ . So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{671 - 509}{115}$

$Z = 1.41$

Idonna's standardized score is 1.41.

Jonathan took the ACT and scored 24 on the mathematics portion. ACT math scores for the same year were Normally distributed with mean 21.1 and standard deviation 5.3 .

This means that his standardized score is Z when $X = 24, \mu = 21.1, \sigma = 5.3$