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In algebra, solving inequalities is similar to solving equations, but with a crucial difference: inequalities have a range of values that satisfy the condition rather than a single solution. The exercise at hand involves checking whether a given number satisfies an inequality. Given the inequality, our job is to determine if the chosen value for the variable makes the inequality true. The steps in our example involved substituting the value into the inequality and simplifying both sides to see if the inequality holds. This process helps us understand whether the solution interval includes the given number. Inequalities use symbols such as \( < \), \( > \), \( \leq \), and \( \geq \) to denote the relationship between expressions.

The substitution method is widely used to check if a particular value satisfies an equation or inequality. It involves replacing the variable with the given number. For example:

Suppose we have the inequality \( 5 - x < 4 - 2x \), and we want to check if \( x = -1 \) makes this true.

• First, substitute \( x \) with \( -1 \) in the inequality:
• \( 5 - (-1) < 4 - 2(-1) \)

By substituting, we transform the original inequality into a numerical comparison, making it easier to verify its truth.

Simplification helps us transform complicated expressions into simpler ones. After substituting the variable, we simplify both sides to compare them.

• On the left side of our example inequality \( 5 - (-1) = 5 + 1 = 6 \).
• On the right side, \( 4 - 2(-1) = 4 + 2 = 6 \).

Now, our simplified inequality reads \( 6 < 6 \). Simplifying ensures we correctly assess the comparison.
In this case, \( 6 < 6 \) is false, so the inequality is not satisfied for \( x = -1 \). This step ensures our calculations are accurate, allowing for a correct conclusion. Understanding this is crucial for solving inequalities and helps ensure the solution is valid.