Section 1
Properties of Inequalities
In Exercises $1-4,$ make the given changes in the indicated examples of this section and then perform the indicated operations.In Example $2,$ in the first paragraph, change the $ > $ to $ < $ and then complete the meaning of the resulting inequality as in the first sentence. Rewrite the meaning as in the second line.
In Exercises $1-4,$ make the given changes in the indicated examples of this section and then perform the indicated operations.In Example $4(\mathrm{b}),$ change $\leq$ to $>$ and then graph the resulting inequality.
In Exercises $1-4,$ make the given changes in the indicated examples of this section and then perform the indicated operations.In Example $7,$ change the inequality to $-2>-4$ and then perform the two operations shown in color.
In Exercises $1-4,$ make the given changes in the indicated examples of this section and then perform the indicated operations.In Example $9(b),$ change the -1 to -3 and the 3 to 1 and then write the two forms in which an inequality represents the statement.
In Exercises $5-12,$ for the inequality $4<9,$ state the inequality that results when the given operations are performed on both members.Add 5
In Exercises $5-12,$ for the inequality $4<9,$ state the inequality that results when the given operations are performed on both members.Subtract 16
In Exercises $5-12,$ for the inequality $4<9,$ state the inequality that results when the given operations are performed on both members.Multiply by 4
In Exercises $5-12,$ for the inequality $4<9,$ state the inequality that results when the given operations are performed on both members.Multiply by -2
In Exercises $5-12,$ for the inequality $4<9,$ state the inequality that results when the given operations are performed on both members.Divide by -1
In Exercises $5-12,$ for the inequality $4<9,$ state the inequality that results when the given operations are performed on both members.Divide by 0.5
In Exercises $5-12,$ for the inequality $4<9,$ state the inequality that results when the given operations are performed on both members.Square both.
In Exercises $5-12,$ for the inequality $4<9,$ state the inequality that results when the given operations are performed on both members.Take square roots.
In Exercises $13-24,$ give the inequalities equivalent to the following statements about the number $x .$Greater than -2
In Exercises $13-24,$ give the inequalities equivalent to the following statements about the number $x .$Less than 0.7
In Exercises $13-24,$ give the inequalities equivalent to the following statements about the number $x .$Less than or equal to 38
In Exercises $13-24,$ give the inequalities equivalent to the following statements about the number $x .$Greater than or equal to -6
In Exercises $13-24,$ give the inequalities equivalent to the following statements about the number $x .$Greater than 1 and less than 7
In Exercises $13-24,$ give the inequalities equivalent to the following statements about the number $x .$Greater than or equal to -200 and less than 650
In Exercises $13-24,$ give the inequalities equivalent to the following statements about the number $x .$Less than $-5,$ or greater than or equal to 3
In Exercises $13-24,$ give the inequalities equivalent to the following statements about the number $x .$Less than or equal to $8,$ or greater than or equal to 12
In Exercises $13-24,$ give the inequalities equivalent to the following statements about the number $x .$Less than $1,$ or greater than 3 and less than or equal to 5
In Exercises $13-24,$ give the inequalities equivalent to the following statements about the number $x .$Greater than or equal to 0 and less than or equal to $2,$ or greater than 5
In Exercises $13-24,$ give the inequalities equivalent to the following statements about the number $x .$Greater than -2 and less than $2,$ or greater than or equal to 3 and less than 4
In Exercises $13-24,$ give the inequalities equivalent to the following statements about the number $x .$Less than $-4,$ or greater than or equal to 0 and less than or equal to $1,$ or greater than or equal to 5
In Exercises $25-28,$ give verbal statements equivalent to the given inequalities involving the number $x$.$$0<x \leq 9$$
In Exercises $25-28,$ give verbal statements equivalent to the given inequalities involving the number $x$.$$x < 5 \text { or } x > 7$$
In Exercises $25-28,$ give verbal statements equivalent to the given inequalities involving the number $x$.$$x < -10 \text { or } 10 \leq x < 20$$
In Exercises $25-28,$ give verbal statements equivalent to the given inequalities involving the number $x$.$$-1 \leq x < 3 \text { or } 5 < x < 7$$
In Exercises $29-44,$ graph the given inequalities on the number line.$$x < 3$$
In Exercises $29-44,$ graph the given inequalities on the number line.$$x \geq-1$$
In Exercises $29-44,$ graph the given inequalities on the number line.$$x \leq-1 \text { or } x > 0.5$$
In Exercises $29-44,$ graph the given inequalities on the number line.$$x < -300 \text { or } x \geq 0$$
In Exercises $29-44,$ graph the given inequalities on the number line.$$0 \leq x < 5$$
In Exercises $29-44,$ graph the given inequalities on the number line.$$-4 < y < -2$$
In Exercises $29-44,$ graph the given inequalities on the number line.$$x \geq-3 \text { and } x < 5$$
In Exercises $29-44,$ graph the given inequalities on the number line.$x>4$ and $x<3$
In Exercises $29-44,$ graph the given inequalities on the number line.$$x < -1 \text { or } 1 \leq x < 4$$
In Exercises $29-44,$ graph the given inequalities on the number line.$$-3 < x < 0 \text { or } x > 3$$
In Exercises $29-44,$ graph the given inequalities on the number line.$$-3 < x < -1 \text { or } 0.5 < x \leq 3$$
In Exercises $29-44,$ graph the given inequalities on the number line.$$x \leq 4 \text { or } x > -4$$
In Exercises $29-44,$ graph the given inequalities on the number line.$$t \leq-5 \text { and } t \geq-5$$
In Exercises $29-44,$ graph the given inequalities on the number line.$$x < 1 \text { or } 1 < x \leq 4$$
In Exercises $29-44,$ graph the given inequalities on the number line.$$(x \leq 5 \text { or } x \geq 8) \text { and }(3 < x < 10)$$
In Exercises $29-44,$ graph the given inequalities on the number line.$(x < 7 \text { and } x > 2)$ or $(x \geq 10 \text { or } x < 1)$
In Exercises $45-48,$ answer the given questions about the inequality $0 < a < b$.Is $a^{2} < b^{2}$ a conditional inequality or an absolute inequality?
In Exercises $45-48,$ answer the given questions about the inequality $0 < a < b$.Is $|a-b| < b-a ?$
In Exercises $45-48,$ answer the given questions about the inequality $0 < a < b$.If each member of the inequality $2 > 1$ is multiplied by $a-b,$ is the result $2(a-b) > (a-b) ?$
In Exercises $45-48,$ answer the given questions about the inequality $0 < a < b$.What is wrong with the following sequence of steps?$a < b, a b < b^{2}, a b-b^{2} < 0, b(a-b) < 0, b < 0$
In Exercises $49-52,$ solve the given problems.Write the relationship between $(|x|+|y|)$ and $|x+y|$ if $x > 0$ and $y < 0$
In Exercises $49-52,$ solve the given problems.Write the relationship between $|x y|$ and $|x||y|$ if $x > 0$ and $y < 0$.
In Exercises $49-52,$ solve the given problems.Explain the error in the following "proof" that $3 < 2$ (1) $1 / 8 < 1 / 4 \quad$ (2) $0.5^{3} < 0.5^{2} \quad$ (3) $\log 0.5^{3} < \log 0.5^{2}$ (4) $3 \log 0.5 < 2 \log 0.5 \quad(5) 3 < 2$
In Exercises $49-52,$ solve the given problems.If $x \neq y,$ show that $x^{2}+y^{2} > 2 x y$
In Exercises $53-62,$ some applications of inequalities are shown.The length $L$ and width $w$ (in yd) of a rectangular soccer field should satisfy the inequalities $110 \leq L \leq 120$ and $70 \leq w \leq 80$ Express the possible diagonal lengths $d$ as an inequality.
In Exercises $53-62,$ some applications of inequalities are shown.A breakfast cereal company guarantees the calorie count shown for each serving is accurate within $5 \% .$ If the package shows a serving has 200 cal, write an inequality for the possible calorie counts.
In Exercises $53-62,$ some applications of inequalities are shown.An electron microscope can magnify an object from 2000 times to 1,000,000 times. Assuming these values are exact, express these magnifications $M$ as an inequality and graph them.
In Exercises $53-62,$ some applications of inequalities are shown.A busy person glances at a digital clock that shows $9: 36 .$ Another glance a short time later shows the clock at $9: 44 .$ Express the amount of time $t$ (in min) that could have elapsed between glances by use of inequalities. Graph these values of $t$.
In Exercises $53-62,$ some applications of inequalities are shown.An Earth satellite put into orbit near Earth's surface will have an elliptic orbit if its velocity $v$ is between $18,000 \mathrm{mi} / \mathrm{h}$ and $25,000 \mathrm{mi} / \mathrm{h} .$ Write this as an inequality and graph these values of $v$.
In Exercises $53-62,$ some applications of inequalities are shown.Fossils found in Jurassic rocks indicate that dinosaurs flourished during the Jurassic geological period, 140 MY (million years ago) to 200 MY. Write this as an inequality, with $t$ representing past time. Graph the values of $t$.
In Exercises $53-62,$ some applications of inequalities are shown.A DVD player spins at 1530 r/min at the innermost edge and gradually slows to a rate of $630 \mathrm{r} / \mathrm{min}$ at the outer edge. Use an inequality to express the angular velocity $\omega$ of the DVD player.
In Exercises $53-62,$ some applications of inequalities are shown.The velocity $v$ of an ultrasound wave in soft human tissue may be represented as $1550 \pm 60 \mathrm{m} / \mathrm{s},$ where the $\pm 60 \mathrm{m} / \mathrm{s}$ gives the possible variation in the velocity. Express the possible velocities by an inequality.
In Exercises $53-62,$ some applications of inequalities are shown.A driver using the Google Maps app finds that it is 300 mi to her destination. If her speed always stays between $50 \mathrm{mi} / \mathrm{h}$ and $60 \mathrm{mi} / \mathrm{h}$ use an inequality to express the required time $t$ for the trip.
In Exercises $53-62,$ some applications of inequalities are shown.If the current from the source in Example 12 is $i=5 \cos 4 \pi t$ and the diode allows only negative current to flow, write the inequalities and draw the graph for the current in the circuit as a function of time for $0 \leq t \leq 1 s$.