Problem 21 - Prealgebra

Question

Solve each inequality. Check your answer. $$-8>h-1$$

Step-by-Step Solution

Answer

The solution is \(h < -7\).

Step 1: Isolate Variable

To solve the inequality \(-8 > h - 1\), we first need to isolate the variable \(h\). This can be done by adding 1 to both sides of the inequality, resulting in the following expression: \(-8 + 1 > h\).

Step 2: Simplify Both Sides

After adding 1 to both sides, we simplify the expression: \(-8 + 1 = -7\). So the inequality becomes \(-7 > h\) or equivalently, \(h < -7\).

Step 3: Check the Solution

To check if \(h < -7\) satisfies the original inequality \(-8 > h - 1\), substitute a value less than -7 into the inequality. For instance, if \(h = -8\), then \(-8 > -8 - 1\) simplifies to \(-8 > -9\), which is true. Thus, our solution is correct.

Key Concepts

Solving Inequalities

Inequalities are statements about the relative size of two values, often involving an unknown variable you need to solve for. For example, in the inequality \(-8 > h - 1\), you are asked to find values of \(h\) that make this statement true. Solving inequalities involves altering them while keeping the inequality valid. This is similar to solving equations, but with one crucial difference: when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.

Here’s a step-by-step approach to solve \(-8 > h - 1\):

• Start by performing operations to isolate the variable, \(h\), on one side of the inequality.
• Add or subtract terms to both sides as needed, just as you would in an equation.
• Only when multiplying or dividing by a negative, remember to flip the inequality sign.

In our example, we added 1 to both sides: \(-8 + 1 > h\), simplifying it to \(-7 > h\) or \(h < -7\). Once simplified, be sure to verify your solution.

Checking Solutions

Once you've found a solution to an inequality, checking its correctness will reinforce your understanding and provide assurance.
Checking involves substituting a sample value back into the original inequality to ensure it results in a true statement. For the inequality \(h < -7\), choose a number smaller than \(-7\) like \(-8\). By substituting, we test if \(-8 > h - 1\) holds: so substitute \(h\) with \(-8\) making it \(-8 > -8 - 1\).

Simplifying \(-8 > -9\) shows a true statement, verifying our solution.

• Pick a test value for \(h\) that should satisfy the inequality.
• Substitute back into the original inequality equation.
• Simplify and see if the inequality holds true.

Remember, if your test value satisfies the original inequality, your solution set is likely correct. Don’t forget to verify multiple values if unsure!

Variable Isolation

Isolating the variable is often the first and key task when working with inequalities. It's the process of rearranging your inequality so that the unknown variable appears on one side, and the numbers are on the other. This step is crucial because it simplifies the inequality, making it easier to interpret.
Here's how to isolate variables effectively:

• Identify operations affecting the variable, like subtraction or division, and perform the inverse operation on both sides.
• In \(-8 > h - 1\), adding 1 to both sides was needed to move the \(-1\) away from \(h\).
• After doing so, you simplify the expression, \(-8 + 1 = -7\).

Once isolated, it becomes straightforward to see what range of values the variable can take. In conclusion, if you consistently work towards isolating the variable correctly, you'll have more success in solving inequalities accurately.