Problem 38 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. $$ |5-x|>3 $$

Step-by-Step Solution

Answer

x \geq -3; [-3, \infty)

Step 1 - Distribute the Left Side

Distribute the \right side of the inequality: \(-2(x + 4) \rightarrow -2x - 8\)

Step 2 - Write Inequality

Rewrite the inequality with the distributed term: \(-2x - 8 \leq 6x + 16\)

Step 3 - Combine Like Terms

Add 2x to both sides to move the x terms to one side: \(-8 \leq 8x + 16\)

Step 4 - Isolate x-term

Subtract 16 from both sides to isolate the 8x term: \(-8 - 16 \leq 8x\) simplifies to \(-24 \leq 8x\)

Step 5 - Solve for x

Divide both sides by 8: \(\frac{-24}{8} \leq x\) simplifies to \(-3 \leq x\), or equivalently \(x \geq -3\)

Step 6 - Graph the Solution Set

Represent the solution on a number line with a closed circle at \(-3\) and a shaded line extending to the right, indicating that \(x\) can be any number greater than or equal to \(-3\).

Step 7 - Write Solution in Interval Notation

Express the solution in interval notation: \([-3, \infty)\)

Key Concepts

graphing inequalities

When working with inequalities, graphing the solution helps visualize which values of the variable satisfy the inequality. For our problem \(-3 \leq x\), we're looking for all values of \(x\) that are greater than or equal to \(-3\). To graph this, follow these simple steps:

• Draw a number line
• Put a closed circle on \(-3\) (closed because the inequality includes \(-3\) as a solution
• Shade the line to the right of \(-3\), extending forever to the right, indicating \(x\) can be any number greater than \(-3\)

This visual aid makes it easy to see which numbers satisfy the inequality.

interval notation

Interval notation is a way of writing subsets of the real number line. For the inequality \(x \geq -3\), we use interval notation to succinctly express the set of all real numbers that make the inequality true. Here’s how to express it:

• The square bracket \([-3\) indicates that \(-3\) is included in the interval (closed interval)
• The parenthesis \(\backslashinfty\) indicates that the interval extends infinitely to the right
• Hence, \([-3, \backslashinfty)\) is the interval notation for \(-3 \leq x\)

This method is compact and easy to read, especially for complex intervals.

distributing coefficients

Distributing coefficients is an important step in solving inequalities. It involves multiplying each term within the parentheses by the coefficient outside. For our problem, \(-2(x + 4) \leq 6x + 16\), we distributed \(-2\) as follows:

• Multiply \(-2\) by \(x\): \(-2x\)
• Multiply \(-2\) by \(4\): \(-8\)

Resulting in: \(-2x - 8\). Properly distributing coefficients helps simplify the expression and make it easier to isolate the variable. Always ensure you multiply each term inside the brackets to avoid mistakes.