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Chapter 17: Problem 11

Solve the given inequalities. Graph each solution. $$12-2 y>16$$

Short Answer

Expert verified
The solution is \(y < -2\), and it's graphed with an open circle at \(-2\) and shading to the left.

Step by step solution

01

Isolate the variable term

Start by isolating the term with the variable on one side of the inequality. Given the inequality \(12 - 2y > 16\), subtract 12 from both sides to get \(-2y > 4\).
02

Solve for the variable

Now, solve for \(y\) by dividing both sides by \(-2\). Remember that when you divide or multiply an inequality by a negative number, the inequality sign flips. Thus, \(-2y > 4\) becomes \(y < -2\).
03

Graph the solution

To graph the solution \(y < -2\), draw a number line. Place an open circle at \(-2\) to indicate that \(-2\) is not included in the solution. Shade the line to the left of \(-2\) to represent all numbers less than \(-2\).

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Key Concepts

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

Solving Inequalities
Inequalities involve expressions where two values are compared using inequality signs such as greater than (>), less than (<), greater than or equal to (鈮�), or less than or equal to (鈮�). To solve inequalities, the goal is to isolate the variable on one side while keeping the inequality true. This process is similar to solving equations, but there is one important additional rule.
  • When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This step is vital to maintain the inequality's truth.
  • For example, in the inequality given: \(12 - 2y > 16\), first subtract 12 from both sides to isolate the term with the variable, leading to \(-2y > 4\).
  • To solve for \(y\), divide both sides by \(-2\). Since you're dividing by a negative, flip the sign to obtain \(y < -2\).
Be attentive to these transformations, checking each step to ensure you maintain the inequality conditions correctly.
Inequality Graphing
Graphing inequalities allows you to vividly represent solutions by showing all possible values satisfying the inequality condition. Once you have solved an inequality and found the range of solutions, you graph these on a number line.
  • For the solution \(y < -2\), use an open circle at \(-2\). The open circle signifies that \(-2\) itself is not part of the solution set.
  • Shade to the left because the solution includes all values less than \(-2\).
  • By visually representing \(y < -2\) on the number line, you clarify which numbers meet the inequality condition.
Inequality graphing serves as a quick visual cue, perfectly depicting ranges in solutions without needing to list them all.
Number Line Representation
A number line representation of an inequality solution makes it easier to understand and visualize the set of possible values. It helps in quickly interpreting the range an inequality covers.
  • The number line should be neatly labeled with numbers, typically including at least the critical point(s) involved in your inequality.
  • For \(y < -2\), an open circle on \(-2\) symbolizes exclusion from the solution set. The openness indicates that while \(-2\) is significant, it isn't included.
  • Shading or elongating a line towards negative infinity (to the left) demonstrates the endlessness of possible values satisfying the condition \(y < -2\).
By regularly practicing number line representation, you enhance your understanding and ability to swiftly solve and interpret inequalities.

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Most popular questions from this chapter

Find the indicated maximum and minimum values by the linear programming method of this section. For Exercises \(5-16,\) the constraints are shown below the objective function. Graphing the constraints of a linear programming problem shows the consecutive vertices of the region of feasible points to be \((1,3),(8,0),(9,7),(5,8),(0,6),\) and \((1,3) .\) What are the maximum and minimum values of the objective function \(F=2 x+5 y\) in this region?

graph the given inequalities on the number line. $$x \geq-3 \text { and } x < 5$$

Draw a sketch of the graph of the region in which the points satisfy the given system of inequalities. $$\begin{aligned} &y>x^{2}\\\ &y

Draw a sketch of the graph of the region in which the points satisfy the given system of inequalities. $$\begin{aligned} &2 y>x\\\ &y>1-x \end{aligned}$$

Solve the given inequalities. Graph each solution. $$-4 t>12$$
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