Problem 16 - Precalculus Student Solutions Manual 5th

Question

Solve each inequality. Write each solution set in interval notation. $$-4 x+3 \geq-2+x$$

Step-by-Step Solution

Answer

The solution set is \((-\infty, 1]\).

Step 1 - Move Variables to One Side

To start, move all the variable terms to one side of the inequality. Subtract \(x\) from both sides: \(-4x + 3 - x \geq -2 + x - x\) simplifies to \(-5x + 3 \geq -2\).

Step 2 - Move Constants to the Other Side

Next, isolate the variable term by moving the constant to the other side. Subtract 3 from both sides: \(-5x + 3 - 3 \geq -2 - 3\) simplifies to \(-5x \geq -5\).

Step 3 - Solve for x

Now, solve for \(x\) by dividing both sides by -5. Remember to reverse the inequality sign when dividing by a negative number: \(-5x / -5 \leq -5 / -5\), which simplifies to \(x \leq 1\).

Step 4 - Express in Interval Notation

Finally, write the solution set in interval notation. Since \(x\) can be less than or equal to 1, the interval notation is \((-\infty, 1]\).

Key Concepts

Interval Notation

Interval notation is a way of writing the set of all solutions to an inequality. It helps to visualize the range of values that satisfy the inequality.
For example, if you have an inequality such as \(x \leq 1\), it means that x can take any value up to and including 1.
In interval notation, you write this as \((-\infty, 1]\).
Here's a quick guide:

• Parentheses \(( )\) mean that the endpoint is not included.
• Square brackets \([ ]\) mean that the endpoint is included.
• \((-\infty, a)\) means all values less than a.
• \([(a, +\infty))\) means all values greater than a.

This notation is very handy for expressing the results of inequalities in a compressed form.
Let's remember our exercise. We found that \(x\leq 1\), which means our interval begins from \(-\infty\) and includes values up to 1. Therefore, the interval notation for this solution is \((-\infty, 1]\).

Isolating Variable

Isolating the variable is the key step in solving any inequality or equation.
To isolate the variable means to get the variable alone on one side of the inequality or equation.
This involves a series of inverse operations to move other terms to the opposite side.
For our exercise, we started with the inequality \(-4x + 3 \geq -2 + x\).
We first moved the x term on the right-hand side to the left-hand side by subtracting x from both sides.
This gave us \-4x + 3 - x \geq -2 + x - x.\ We simplified this to \-5x + 3 \geq -2.\ After this, we focused on the constant term 3. By subtracting 3 from both sides, we simplified it to \-5x \geq -5.\ Inverse operations (adding, subtracting, multiplying, or dividing) are essentially the tools we use to isolate the variable. The critical point to remember is that when you multiply or divide by a negative number, you must reverse the inequality sign. This step is essential to accurately solving the inequality.

Solving Linear Inequalities

Solving linear inequalities follows steps similar to solving linear equations, but with extra attention to the inequality sign.
The process includes:

• Moving variable terms to one side.
• Moving constant terms to the other side.
• Simplifying both sides where possible.
• Dividing or multiplying to isolate the variable.

In our example, the initial inequality was \(-4x + 3 \geq -2 + x\).
We first moved variables by subtracting x from both sides:
\(-4x + 3 - x \geq -2 + x - x\)
which simplified to \(-5x + 3 \geq -2\).
Next, we moved the constant by subtracting 3 from both sides:
\(-5x + 3 - 3 \geq -2 - 3\) resulting in \(-5x \geq -5\).
Finally, we divided by -5 and reversed the inequality sign:
\(-5x / -5 \leq -5 / -5\) to get
\(x \leq 1\).
Always remember to flip the inequality sign when dividing by a negative number.
This attention to detail ensures the correct solution for linear inequalities.