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Solving inequalities is similar to solving equations, but instead of looking for a specific value, you are finding a range of values that satisfy the given statement. You want to isolate the variable on one side of the inequality sign. For example, in the inequality \( t > -7 \), you simply read this as 鈥淺( t \) is greater than -7.鈥� This tells us that any value of \( t \) greater than -7 will satisfy this inequality.

Remember, an inequality sign can be one of the following:

• Greater than (\( > \))
• Less than (\( < \))
• Greater than or equal to (\( \ge \))
• Less than or equal to (\( \le \))

By solving inequalities accurately, you can determine the correct range of values for the variable.

In the context of inequalities, equivalence refers to different expressions that represent the same range of values. For inequalities to be equivalent, they must not contradict each other. For example, the inequalities \( t > -7 \) and \( 7 < -t \) are given. To verify if they are equivalent, we transform \( 7 < -t \).

We rewrite \( 7 < -t \) by multiplying both sides by -1. This changes the inequality to \( -7 > t \) or equivalently \( t < -7 \). Now we have two inequalities: \( t > -7 \) and \( t < -7 \). These inequalities are contradictory because \( t \) cannot be both greater than and less than -7 simultaneously.

Therefore, these two inequalities are not equivalent since their solution sets do not overlap or match.

When solving inequalities, if you multiply or divide both sides of the inequality by a negative number, you must reverse the inequality sign. This is a key rule to remember. For example, starting with \( 7 < -t \), you multiply both sides by -1.

This step changes the inequality from \( 7 < -t \) to \( -7 > t \), which simplifies to \( t < -7 \).

• Original: \( 7 < -t \)
• Multiply both sides by -1: \( -7 > t \)
• Simplified: \( t < -7 \)

Hence, multiplying or dividing by negative values reverses the initial inequality sign to preserve the truth of the statement. Ignoring this crucial step when operating on inequalities will lead to incorrect solutions and misunderstanding.