a. t + 6 ≤ 2 + 3t

The solution to this inequality can be determined by first isolating the variable on one side of the equation and then solving for any possible values that satisfy it. To do so we start by subtracting ‘3t’ from both sides of the equation:

t + 6 – 3t ≤ 2

-2t ≤ -4

We then divide both sides by ‘-2’ in order to solve for ‘t’:

-2t/-2 ≤ -4/-2

t ≥ 2

This means that any value greater than or equal to two will satisfy our inequalities; thus we can say that our answer is:

[2, ∞) which can be represented geometrically on a number line as such:

[image]

b. 3(2 – 3x) > 4(1 – 4x)

To solve this inequality, we again begin by isolating the x variable on one side of the equation and then finding out what values can make it true. To do so we start off by multiplying both sides by ‘-1/16’ in order to get rid of fractions:

−3\times \frac{1}{16} (2 − 3x) > −4 \times \frac{1}{16} (1 − 4x)

\frac{3}{8} − \frac{9}{16} x > \frac{4}{8} − \frac {16}{16} x

Then we add ‘9/16’ to both sides and divide them each with a common denominator in order to obtain our answer:

⇒

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