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Chapter 7: Problem 16

Solve each inequality and graph the solution on a number line. a. \(-2<6 x+8\) (a) b. \(3(2-x)+4 \geq 13\) (a) c. \(-0.5 \geq-1.5 x+2(x-4)\)

Short Answer

Expert verified
a) \(x > -\frac{5}{3}\); b) \(x \leq -1\); c) \(x \leq 15\).

Step by step solution

01

Simplify and Isolate for a

Start with the inequality \(-2<6x+8\). Subtract 8 from both sides to get \(-10<6x\). Then, divide every term by 6 to isolate \(x\). This results in \(x > -\frac{5}{3}\).
02

Graph Solution for a

Draw a number line and mark \(-\frac{5}{3}\). Since the inequality is \(x > -\frac{5}{3}\), use an open circle to indicate that \(-\frac{5}{3}\) is not included, and shade the line to the right to represent all numbers greater than \(-\frac{5}{3}\).
03

Distribute and Isolate for b

Start with the inequality \(3(2-x)+4 \geq 13\). Distribute the 3 to get \(6-3x+4 \geq 13\), which simplifies to \(10-3x \geq 13\). Subtract 10 from both sides to get \(-3x \geq 3\). Divide by -3, and remember to reverse the inequality sign to get \(x \leq -1\).
04

Graph Solution for b

On a number line, mark \(-1\) with a closed circle since the inequality includes \(-1\). Shade the line to the left to indicate all numbers less than or equal to \(-1\).
05

Simplify and Isolate for c

Start with the inequality \(-0.5 \geq -1.5x + 2(x-4)\). First, distribute 2 to get \(-0.5 \geq -1.5x + 2x - 8\). Combine like terms to simplify it to \(-0.5 \geq 0.5x - 8\). Add 8 to both sides to yield \(7.5 \geq 0.5x\). Divide each side by 0.5 to obtain the solution \(x \leq 15\).
06

Graph Solution for c

On a number line, mark 15 with a closed circle since \(x\) includes 15. Shade the line to the left to represent all numbers less than or equal to 15.

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Key Concepts

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

Graphing Solutions
When solving inequalities, graphing the solution gives us a visual representation of all possible solutions on the number line. It's an essential part of understanding inequalities as it helps to interpret the solution set.

To graph solutions:
  • Identify the critical point鈥攖his is the number where the inequality changes.
  • Determine whether to use an open or closed circle:
    • Use an open circle for "less than" (<) or "greater than" (>) which means the point is not included in the solution.
    • Use a closed circle for "less than or equal to" (鈮�) or "greater than or equal to" (鈮�) which means the point is included.
  • Shade the portion of the number line that represents all the numbers that satisfy the inequality:
    • Shade right for "greater than" (>) or "greater than or equal to" (鈮�).
    • Shade left for "less than" (<) or "less than or equal to" (鈮�).
With practice, graphing solutions becomes intuitive and is a helpful tool for verifying your calculated solutions.
Number Line
A number line is a visual representation of numbers laid out on a straight path where equal intervals indicate the value of integers and other real numbers. It's a fundamental concept in mathematics and is extremely useful when working with inequalities.

Here's how to use a number line effectively with inequalities:
  • Position values on the line depending on whether they're positive or negative.
  • Mark 'important' numbers from your inequality solution clearly, typically with circles.
  • Use direction (left or right) to demonstrate the range of solutions.
Distributive Property
The distributive property is a powerful algebraic tool used to simplify expressions and make equations easier to solve. It allows you to multiply a single term by each term within a parenthesis.

For example, in the expression:

\[a(b + c) = ab + ac\]

This property makes complex algebraic operations more manageable by breaking them down into simpler components. Often encountered in solving inequalities, like in the example:
  • Multiply the term outside by every term inside the parenthesis:
    • e.g., \(3(2-x)\) becomes \(6 - 3x\).
  • Combine like terms if applicable to further simplify the equation.
In inequalities, careful use of the distributive property helps to clearly isolate variables and leads to an accurate solution. This simplification is crucial to making the problem easier to graph and understand fully.

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Most popular questions from this chapter

Create this graph on graph paper: When \(x \geq 0\), graph the line \(y=x\). When \(x<0\), graph the line \(y=-x\). What single function has this same graph?

On graph paper, draw a graph that is a function and has these three properties: \- Domain of \(x\)-values satisfying \(-3 \leq x \leq 5\) \- Range of \(y\)-values satisfying \(-4 \leq y \leq 4\) Includes the points \((-2,3)\) and \((3,-2)\) (a)

Explain why the equation \(x^{2}=-4\) has no solutions.

Use function notation to write the equation of a line through each pair of points. a. \((0,5)\) and \((1,12)\) (a) b. \((1,5)\) and \((2,12)\)

For each relationship, identify the independent variable and the dependent variable. a. the weight of your dog and the reading on the scale b. the amount of time you spend in an airplane and the distance between your departure and your destination c. the number of times you dip a wick into hot wax and the diameter of a handmade candle
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