How Do You Know When an Inequality Solution Has No Solution

Learning Objectives

Past the end of this department, you will be able to:

• Solve quadratic inequalities graphically
• Solve quadratic inequalities algebraically

Be Prepared 9.22

Earlier you go started, take this readiness quiz.

Solve: $2x-\mathrm{iii}=0.$
If you missed this problem, review Case 2.2.

Be Prepared 9.23

Solve: $2y^2+y=15$ .
If you missed this trouble, review Example 6.45.

Be Prepared 9.24

Solve $\frac{1}{{\mathrm{ten}}^{\mathrm{two}}+2x-\mathrm{eight}}>0$
If you missed this problem, review Instance 7.56.

We have learned how to solve linear inequalities and rational inequalities previously. Some of the techniques we used to solve them were the same and some were different.

We volition at present acquire to solve inequalities that have a quadratic expression. We volition utilize some of the techniques from solving linear and rational inequalities besides equally quadratic equations.

We will solve quadratic inequalities two ways—both graphically and algebraically.

Solve Quadratic Inequalities Graphically

A quadratic equation is in standard form when written as ax ² + bx + c = 0. If we replace the equal sign with an inequality sign, we have a quadratic inequality in standard class.

Quadratic Inequality

A quadratic inequality is an inequality that contains a quadratic expression.

The standard form of a quadratic inequality is written:

$$\begin{array}{cccccc}a{x}^{\mathrm{two}}+bx+c<0\hfill & & & & & a{x}^2+bx+c\le 0\hfill \\ a{x}^{\mathrm{two}}+b\mathrm{10}+c>0\hfill & & & & & a{x}^{\mathrm{ii}}+bx+c\ge 0\hfill \end{array}$$

The graph of a quadratic function f(ten) = ax ² + bx + c = 0 is a parabola. When we ask when is ax ⁱⁱ + bx + c < 0, nosotros are asking when is f(x) < 0. Nosotros want to know when the parabola is below the x-centrality.

When nosotros ask when is ax ² + bx + c > 0, we are asking when is f(x) > 0. Nosotros want to know when the parabola is above the y-axis.

Case 9.64

How to Solve a Quadratic Inequality Graphically

Solve $x^2-6\mathrm{ten}+8<0$ graphically. Write the solution in interval note.

Try Information technology 9.127

ⓐ Solve $x^{\mathrm{two}}+2x-8<0$ graphically and ⓑ write the solution in interval notation.

Attempt It 9.128

ⓐ Solve $x^2-8x+12\ge 0$ graphically and ⓑ write the solution in interval note.

We list the steps to accept to solve a quadratic inequality graphically.

How To

Solve a quadratic inequality graphically.

1. Stride 1. Write the quadratic inequality in standard class.
2. Stride 2. Graph the role $f\left(x\right)=a{x}^{\mathrm{ii}}+b\mathrm{ten}+c.$
3. Step 3. Determine the solution from the graph.

In the concluding example, the parabola opened upwardly and in the next example, it opens downwards. In both cases, we are looking for the role of the parabola that is below the 10-axis only notation how the position of the parabola affects the solution.

Case 9.65

Solve $\text{−}{\mathrm{10}}^2-8x-12\le 0$ graphically. Write the solution in interval note.

Try It nine.129

ⓐ Solve $\text{−}{\mathrm{ten}}^2-6x-5>0$ graphically and ⓑ write the solution in interval annotation.

Try It 9.130

ⓐ Solve $\text{−}{x}^{\mathrm{ii}}+\mathrm{x}\mathrm{10}-16\le 0$ graphically and ⓑ write the solution in interval notation.

Solve Quadratic Inequalities Algebraically

The algebraic method we will use is very similar to the method we used to solve rational inequalities. We volition notice the critical points for the inequality, which will be the solutions to the related quadratic equation. Remember a polynomial expression can modify signs only where the expression is nil.

We volition use the disquisitional points to divide the number line into intervals and then make up one's mind whether the quadratic expression willl be postive or negative in the interval. Nosotros and so make up one's mind the solution for the inequality.

Example nine.66

How To Solve Quadratic Inequalities Algebraically

Solve ${\mathrm{ten}}^2-x-12\ge 0$ algebraically. Write the solution in interval notation.

Effort Information technology ix.131

Solve ${\mathrm{10}}^2+2x-\mathrm{viii}\ge 0$ algebraically. Write the solution in interval notation.

Endeavour It 9.132

Solve $x^2-2x-\mathrm{fifteen}\le 0$ algebraically. Write the solution in interval notation.

In this instance, since the expression ${\mathrm{ten}}^2-x-12$ factors nicely, we can also find the sign in each interval much like nosotros did when we solved rational inequalities. Nosotros find the sign of each of the factors, and then the sign of the product. Our number line would like this:

The result is the aforementioned as nosotros found using the other method.

We summarize the steps here.

How To

Solve a quadratic inequality algebraically.

1. Step 1. Write the quadratic inequality in standard form.
2. Stride 2. Determine the disquisitional points—the solutions to the related quadratic equation.
3. Step three. Employ the critical points to split up the number line into intervals.
4. Step iv. Above the number line testify the sign of each quadratic expression using test points from each interval substituted into the original inequality.
5. Footstep 5. Make up one's mind the intervals where the inequality is right. Write the solution in interval notation.

Example 9.67

Solve ${-\mathrm{ten}}^2+6x-7\ge 0$ algebraically. Write the solution in interval notation.

Try It 9.133

Solve $\text{−}{\mathrm{10}}^{\mathrm{two}}+2\mathrm{10}+\mathrm{i}\ge 0$ algebraically. Write the solution in interval notation.

Endeavor It 9.134

Solve $\text{−}{x}^{\mathrm{two}}+8x-14<0$ algebraically. Write the solution in interval annotation.

The solutions of the quadratic inequalities in each of the previous examples, were either an interval or the union of ii intervals. This resulted from the fact that, in each case we found two solutions to the corresponding quadratic equation ax ⁱⁱ + bx + c = 0. These two solutions then gave us either the ii x-intercepts for the graph or the two critical points to split up the number line into intervals.

This correlates to our previous word of the number and type of solutions to a quadratic equation using the discriminant.

For a quadratic equation of the form ax ² + bx + c = 0, $a\ne 0.$

The last row of the table shows us when the parabolas never intersect the ten-axis. Using the Quadratic Formula to solve the quadratic equation, the radicand is a negative. We go 2 complex solutions.

In the next example, the quadratic inequality solutions will upshot from the solution of the quadratic equation being complex.

Example ix.68

Solve, writing any solution in interval notation:

ⓐ ${\mathrm{10}}^{\mathrm{ii}}-\mathrm{three}x+4>0$ ⓑ $x^2-\mathrm{iii}x+\mathrm{iv}\le 0$

Try It ix.135

Solve and write whatever solution in interval annotation:
ⓐ $\text{−}{x}^2+2x-4\le 0$ ⓑ $\text{−}{x}^2+\mathrm{ii}x-4\ge 0$

Try It 9.136

Solve and write any solution in interval note:
ⓐ $x^{\mathrm{two}}+3x+\mathrm{three}<0$ ⓑ $x^{\mathrm{ii}}+3x+3>0$

Section 9.8 Exercises

Practice Makes Perfect

Solve Quadratic Inequalities Graphically

In the following exercises, ⓐ solve graphically and ⓑ write the solution in interval notation.

363 .

$x^2+6x+\mathrm{five}>0$

364 .

$x^2+\mathrm{iv}x-12<0$

365 .

$x^2+\mathrm{four}x+\mathrm{iii}\le 0$

366 .

${\mathrm{10}}^{\mathrm{ii}}-\mathrm{half-dozen}x+8\ge 0$

367 .

$\text{−}{x}^2-3x+18\le 0$

368 .

$\text{−}{\mathrm{10}}^{\mathrm{two}}+\mathrm{ii}x+24<0$

369 .

$\text{−}{x}^2+x+12\ge 0$

370 .

$\text{−}{x}^{\mathrm{ii}}+2x+15>0$

In the post-obit exercises, solve each inequality algebraically and write any solution in interval notation.

371 .

${\mathrm{ten}}^2+\mathrm{iii}x-4\ge 0$

372 .

${\mathrm{ten}}^{\mathrm{two}}+x-6\le 0$

373 .

$x^{\mathrm{ii}}-\mathrm{seven}x+10<0$

374 .

$x^2-4\mathrm{10}+3>0$

375 .

$x^{\mathrm{two}}+8x>-15$

376 .

$x^{\mathrm{two}}+8\mathrm{10}<-12$

377 .

${\mathrm{10}}^2-4x+\mathrm{ii}\le 0$

378 .

$\text{−}{\mathrm{10}}^{\mathrm{ii}}+\mathrm{viii}\mathrm{10}-\mathrm{xi}<0$

379 .

${\mathrm{10}}^{\mathrm{ii}}-10\mathrm{10}>-19$

380 .

$x^{\mathrm{two}}+6x<-3$

381 .

$\mathrm{-half\; dozen}{\mathrm{ten}}^2+19x-\mathrm{x}\ge 0$

382 .

$\mathrm{-3}{x}^2-4\mathrm{ten}+\mathrm{four}\le 0$

383 .

$\mathrm{-two}{x}^{\mathrm{two}}+7x+4\ge 0$

384 .

$2{\mathrm{10}}^{\mathrm{two}}+5x-12>0$

385 .

$x^2+3x+\mathrm{v}>0$

386 .

${\mathrm{ten}}^2-3x+6\le 0$

387 .

$\text{−}{x}^2+x-7>0$

388 .

$\text{−}{\mathrm{ten}}^2-4x-\mathrm{five}<0$

389 .

$\mathrm{-ii}{x}^2+8x-10<0$

390 .

$\text{−}{\mathrm{10}}^2+2x-\mathrm{seven}\ge 0$

Writing Exercises

391 .

Explain disquisitional points and how they are used to solve quadratic inequalities algebraically.

392 .

Solve $x^{\mathrm{ii}}+\mathrm{ii}x\ge 8$ both graphically and algebraically. Which method practise you prefer, and why?

393 .

Draw the steps needed to solve a quadratic inequality graphically.

394 .

Depict the steps needed to solve a quadratic inequality algebraically.

Cocky Bank check

ⓐ After completing the exercises, apply this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1-10, how would yous rate your mastery of this section in light of your responses on the checklist? How tin can you lot improve this?

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Source: https://openstax.org/books/intermediate-algebra-2e/pages/9-8-solve-quadratic-inequalities

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