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Inequalities can get quite interesting when they form a compound inequality. A compound inequality involves two separate inequalities joined by 'and' or 'or'. In this particular type, both inequalities must be true at the same time. So, they are essentially two conditions that need to be satisfied simultaneously.

In the problem you worked on, the compound inequality was given as:

• \( -1 \leq \frac{1-4t}{3} \leq 1 \)

This means two inequalities are at play:

• \( -1 \leq \frac{1-4t}{3} \) and
• \( \frac{1-4t}{3} \leq 1 \).

Both must be solved and their solutions must overlap. Breaking it down helps in determining a solution that will satisfy both inequalities.Compound inequalities might look complex at first, but once you solve the inequalities individually and piece them together, they become pretty handy in representing a range of possible solutions. It's like ensuring two promises can be true at the same time.

Once you've figured out the solution to an inequality, expressing it often comes next with interval notation. Think of interval notation as a way to succinctly capture a range of numbers that satisfy an inequality condition.

The interval notation uses brackets to show which numbers are included or excluded. For closed boundaries, where the endpoints are included, square brackets \([ \ ]\) are used. For open boundaries, where endpoints are not included, parentheses \(( )\) are used. In the solved inequality:

• \(-0.5 \leq t \leq 1\)

The interval notation becomes

• \([-0.5, 1]\)

This shows a range starting from \(-0.5\) up to \(1\) both inclusive. The interval notation \([-0.5, 1]\) efficiently communicates that any number between \(-0.5\) and \(1\) are solutions to the inequality.

Solving inequalities requires a good grasp of algebraic techniques, but it's not much different from solving regular equations. The key is to perform operations that maintain the truth of the inequality. While similar to solving equations, there are important differences. Here are basic steps when working through inequalities:

• Identify the inequality and its terms
• Clear any fractions or decimals (multiply through by a common denominator)
• Perform algebraic manipulations to isolate the variable (addition, subtraction, multiplication or division)
• Remember **to flip** the inequality sign when multiplying or dividing by a negative number

In the example

• \(-1 \leq \frac{1-4t}{3} \leq 1\)

the fractions were cleared by multiplying through by 3 to simplify solving. Then each half was worked through:

• \(-1 \leq \frac{1-4t}{3} \) became \( -3 \leq 1 - 4t \)
• \(\frac{1-4t}{3} \leq 1\) became \(1 - 4t \leq 3\)

It's crucial to remember that flipping the inequality sign is necessary when dividing or multiplying by a negative. It's these small but critical steps that ensure you solve the inequality correctly and elegantly. Solving inequalities can feel tricky, but following consistent steps makes the process simpler to manage.