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Chapter 11: Problem 1

Solve the following inequalities: (a) \(3+x>7\) (b) \(3 t<6\) (c) \(z-2 \geq 5\) (d) \(\frac{p}{3} \leq 2\)

Short Answer

Expert verified
(a) \(x>4\); (b) \(t<2\); (c) \(z \geq 7\); (d) \(p \leq 6\).

Step by step solution

01

Solve inequality (a)

The given inequality is \(3+x>7\). The goal is to solve for \(x\). To do this, subtract 3 from both sides of the inequality: \(x > 7 - 3\). Simplifying, we find that \(x > 4\).
02

Solve inequality (b)

The given inequality is \(3t<6\). We want to solve for \(t\). Start by dividing both sides of the inequality by 3: \(t < \frac{6}{3}\). Simplifying the right side gives \(t < 2\).
03

Solve inequality (c)

The inequality is \(z-2 \geq 5\). To isolate \(z\), add 2 to both sides of the inequality: \(z \geq 5 + 2\). Simplifying, we get \(z \geq 7\).
04

Solve inequality (d)

The given inequality is \(\frac{p}{3} \leq 2\). To solve for \(p\), multiply both sides by 3: \(p \leq 2 \times 3\). Simplifying, we have \(p \leq 6\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Inequalities
Understanding how to solve inequalities is a crucial skill in algebra. An inequality is a mathematical statement that indicates one quantity is less than, greater than, equal to, or not equal to another quantity. Inequalities use symbols such as \(>\), \(<\), \( \geq \), and \( \leq \). Here's how they work:

  • \(x > 4\) means \(x\) is greater than 4.
  • \(t < 2\) means \(t\) is less than 2.
  • \(z \geq 7\) means \(z\) is greater than or equal to 7.
  • \(p \leq 6\) means \(p\) is less than or equal to 6.
To solve inequalities, you perform operations similar to those used in solving equations. The goal is to isolate the variable on one side. However, keep in mind:

  • If you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. For example, if you have \(-x > 3\), multiplying by \(-1\) gives \(x < -3\).
The steps you follow for solving inequalities often involve simplifying the inequality by using addition, subtraction, multiplication, or division, aiming to get the variable by itself.
Algebraic Expressions
Algebraic expressions form the building blocks of equations and inequalities. They contain variables, constants, and mathematical operators. For example, in the inequality \(3 + x > 7\), \(3 + x\) is an algebraic expression.

  • Variables represent unknown values, such as \(x\), \(t\), \(z\), or \(p\).
  • Constants are fixed numbers like 3, 6, or 2 in our examples.
  • Operators include plus (+), minus (-), multiplication (*), and division (/).
Working with algebraic expressions efficiently involves understanding how to simplify them by combining like terms or factoring expressions. For inequalities, the process involves arranging terms such that the variable is isolated on one side. It鈥檚 essential to pay careful attention to signs and coefficients when rearranging or simplifying.
Mathematical Operations
Mathematical operations are the foundational tools for manipulating algebraic expressions and equations. When solving inequalities, the operations used include addition, subtraction, multiplication, and division.

  • Addition and Subtraction: These operations are used to eliminate constants from algebraic expressions. For instance, in \(3 + x > 7\), you subtract 3 from both sides to get \(x > 4\).
  • Multiplication and Division: These operations help isolate the variable by removing coefficients. For example, \(3t < 6\) becomes \(t < 2\) when both sides are divided by 3.
  • Special Note: If multiplying or dividing by a negative number, remember to reverse the inequality sign. This rule is crucial for maintaining the inequality's truth.
When performing operations on inequalities, it鈥檚 essential to apply the same operation to both sides. This maintains balance, similar to how equations are solved, yet with heightened attention to inequality direction.

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