Study Guide on Inequalities and Algebraic Expressions

Algebraic Expressions and Inequalities

Inequality Problems

Problem 1

• Expression: $<-6(-4-k)$   - This expression is an inequality involving a linear combination of terms with a variable, $k$.

Problem 2

• Expression: $24 + 6k < -6$

Problem 3

• Expression: $-2n - 40 < 5(6 + n) + 7n$   - This inequality combines both $n$ terms and constants, and involves right side simplification.

Problem 4

• Expression: $5b + 2 < -43$   - This inequality involves a variable $b$ and needs isolation to find its solution.

Problem 5

• Expression: $9 + 2s$   - This shows an expression for $s$ which likely needs to be compared against a value.

Problem 6

• Expression: $6(3 - 4r) < 8r - 334$   - This compares quantities involving $r$, needing distribution and collection of like terms.

Problem 7

• Expression: $-4(4 + 7x) + x \, herefore \,-6x + 5$   - Evaluate the left-hand side after distributing $-4$ and combining terms.

Numerical Values and Graph Points

• Graph Details:   - Range of values from -11 through 12.   - Specific integers might be needed to assess the solution or sketch functions on a number line.

Problem 8

• Graph Number Line Placement: Numbers -10 through 5.
• Examine points on the number line such as from -10 to 10 to assess values closer to 0 and how these map to $x$ terms.

Problem 9

• Expression: $5p + 2 > 22$   - This deals with isolating the $p$ variable and checking inequalities.

Problem 10

• Expression: $342 + 6a$
• Evaluate: Need to determine how this expression relates to another value or inequality.

Problem 11

• Expression: $-7(4-x) + 42 - 18 + 7x$   - This represents a combination of constants and $x$, with a focus on resolving and combining to see if inequalities hold.

Conclusion

• The inequalities specified involve simple linear expressions that require algebraic manipulation to isolate variables. Graphing the solutions or determining ranges on the number line may help visualize and solve the problems more clearly.